JEE Advanced Maths PYQ – Conic Sections (Parabola) | Previous Year Questions with Solutions

Q 1 :

Let $P$ be a point on the parabola $y^{2} = 4 a x$ , where $a > 0$ . The normal to the parabola at $P$ meets the $x$ -axis at a point $Q$ . The area of the triangle $P F Q$ , where $F$ is the focus of the parabola, is 120. If the slope $m$ of the normal and $a$ are both positive integers, then the pair $(a, m)$ is [2023]

(2, 3)
(1, 3)
(2, 4)
(3, 4)

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(1)

$Let point P (a m^{2}, - 2 a m)$

$Equation of normal at P (a m^{2}, - 2 a m) is$

$y = m x - 2 a m - a m^{3}$

$Area of ∆ P F Q$

$= \frac{1}{2} (a + a m^{2}) \times 2 a m = 120$

$\Rightarrow a^{2} m (1 + m^{2}) = 120$

$Pair (a, m) = (2, 3) satisfies the above equation.$

Q 2 :

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1 : 3. Then the locus of $P$ is [2011]

$x^{2} = y$
$y^{2} = 2 x$
$y^{2} = x$
$x^{2} = 2 y$

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(3)

$Let A (x, y) = (t^{2}, 2 t) be any point on parabola y^{2} = 4 x .$

$Let P (h, k) divide O A in the ratio 1 : 3$

$∴$ $(h, k) = (\frac{t^{2}}{4}, \frac{2 t}{4}) \Rightarrow h = \frac{t^{2}}{4}, k = \frac{t}{2} \Rightarrow h = k^{2}$

$∴$ $Locus of P (h, k) is x = y^{2}$

Q 3 :

The axis of a parabola is along the line $y = x$ and the distances of its vertex and focus from origin are $\sqrt{2}$ and $2 \sqrt{2}$ respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is [2006]

${(x + y)}^{2} = (x - y - 2)$
${(x - y)}^{2} = (x + y - 2)$
${(x - y)}^{2} = 4 (x + y - 2)$
${(x - y)}^{2} = 8 (x + y - 2)$

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(4)

Since distance of vertex and focus of the parabola from origin is $\sqrt{2}$ and $2 \sqrt{2}$ ,

$∴$ Vertex is (1, 1) and focus is (2, 2), directrix $x + y = 0$

$∴$ Equation of parabola is

${(x - 2)}^{2} + {(y - 2)}^{2} = {(\frac{x + y}{\sqrt{2}})}^{2}$

$\Rightarrow 2 (x^{2} - 4 x + 4) + 2 (y^{2} - 4 y + 4) = x^{2} + y^{2} + 2 x y$

$\Rightarrow x^{2} + y^{2} - 2 x y = 8 (x + y - 2) \Rightarrow {(x - y)}^{2} = 8 (x + y - 2)$

Q 4 :

Tangent to the curve $y = x^{2} + 6$ at a point (1, 7) touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ at a point $Q$ . Then the coordinates of $Q$ are [2005]

$(- 6, - 11)$
$(- 9, - 13)$
$(- 10, - 15)$
$(- 6, - 7)$

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(4)

The given curve is $y = x^{2} + 6$

Equation of tangent at (1, 7) is $\frac{1}{2} (y + 7) = x \cdot 1 + 6 \Rightarrow 2 x - y + 5 = 0 ...(i)$

Since tangent (i) touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ with centre $C (- 8, - 6)$ at Q.

$∴$ Equation of $C Q$ which is perpendicular to (i) is

$y + 6 = - \frac{1}{2} (x + 8) \Rightarrow x + 2 y + 20 = 0 ...(ii)$

On solving equations (i) and (ii), we get the coordinates of $Q$ as $x = - 6, y = - 7$

$∴$ Coordinates of $Q$ is $(- 6, - 7)$

Q 5 :

The angle between the tangents drawn from the point (1, 4) to the parabola $y^{2} = 4 x$ is [2004]

$\frac{π}{6}$
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{2}$

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(3)

If $m$ be the slope of the tangent to the parabola, then its equation is $y = m x + \frac{1}{m}$

Since the tangent passes through (1, 4)

$∴$ $4 = m + \frac{1}{m} \Rightarrow m^{2} - 4 m + 1 = 0$

If the angle between two tangents to the parabola be $θ$ , then

$\tan θ =$ $| \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$ $= \frac{\sqrt{{(m_{1} + m_{2})}^{2} - 4 m_{1} m_{2}}}{1 + m_{1} m_{2}}$

$= \frac{\sqrt{16 - 4}}{1 + 1} = \sqrt{3} \Rightarrow θ = \frac{π}{3}$

Q 6 :

The focal chord to $y^{2} = 16 x$ is tangent to ${(x - 6)}^{2} + y^{2} = 2$ , then the possible values of the slope of this chord are [2003]

${- 1, 1}$
${- 2, 2}$
${- 2, - \frac{1}{2}}$
${2, - \frac{1}{2}}$

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(1)

Given parabola $y^{2} = 16 x$ , its focus = (4, 0). Let $m$ be the slope of focal chord then its equation is

$y = m (x - 4) ...(i)$

But it is given that equation (i) is a tangent to the circle

${(x - 6)}^{2} + y^{2} = 2$ with centre, C(6, 0) and radius $r = \sqrt{2}$

$∴$ Length of perpendicular from (6, 0) to (i) $= \sqrt{2}$

$\Rightarrow \frac{6 m - 4 m}{\sqrt{m^{2} + 1}} = \sqrt{2}$ $\Rightarrow 2 m = \sqrt{2 (m^{2} + 1)}$

$\Rightarrow 2 m^{2} = m^{2} + 1$ $\Rightarrow m = \pm 1$

Q 7 :

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix [2002]

$x = - a$
$x = - \frac{a}{2}$
$x = 0$
$x = \frac{a}{2}$

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(3)

If $(h, k)$ is the mid point of line joining focus $(a, 0)$ and $Q (a t^{2}, 2 a t)$ on parabola then $h = \frac{a + a t^{2}}{2}, k = a t$

Eliminating $t$ , we get $2 h = a + a (\frac{k^{2}}{a^{2}})$

$\Rightarrow k^{2} = a (2 h - a)$ $\Rightarrow$ $k^{2} = 2 a (h - \frac{a}{2})$

$∴$ $Locus of (h, k) is$ $y^{2} = 2 a (x - \frac{a}{2}), which is equation of a parabola$

$whose directrix is (x - \frac{a}{2}) = - \frac{a}{2} \Rightarrow x = 0$

Q 8 :

The equation of the directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is [2001]

$x = - 1$
$x = 1$
$x = - \frac{3}{2}$
$x = \frac{3}{2}$

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(4)

Given equation of the parabola is $y^{2} + 4 y + 4 x + 2 = 0$

$\Rightarrow y^{2} + 4 y + 4 = - 4 x + 2$ $\Rightarrow {(y + 2)}^{2} = - 4 (x - \frac{1}{2})$

It is of the form $Y^{2} = - 4 A X$

Equation of whose directrix is given by $X = A$

$∴$ Equation of required directrix is $x - \frac{1}{2} = 1$ $\Rightarrow x = \frac{3}{2}$

Q 9 :

The equation of the common tangent touching the circle ${(x - 3)}^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the $x$ -axis is [2001]

$\sqrt{3} y = 3 x + 1$
$\sqrt{3} y = - (x + 3)$
$\sqrt{3} y = x + 3$
$\sqrt{3} y = - (3 x + 1)$

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(3)

Let the equation of tangent to the parabola $y^{2} = 4 x$ be

$y = m x + \frac{1}{m},$ where $m$ is the slope of the tangent.

If $y = m x + \frac{1}{m}$ is also tangent to the circle ${(x - 3)}^{2} + y^{2} = 9$ ,

then length of perpendicular to the tangent from centre (3, 0) should be equal to radius 3.

i.e., $\frac{3 m + \frac{1}{m}}{\sqrt{m^{2} + 1}} = 3$ $\Rightarrow 9 m^{2} + \frac{1}{m^{2}} + 6 = 9 m^{2} + 9$ $\Rightarrow m = \pm \frac{1}{\sqrt{3}}$

$∴$ $Tangents are x - \sqrt{3} y + 3 = 0 and x + \sqrt{3} y + 3 = 0$

$out of which x - \sqrt{3} y + 3 = 0 meets the parabola at (3, 2 \sqrt{3}), i.e., above x-axis.$

Q 10 :

If the line $x - 1 = 0$ is the directrix of the parabola $y^{2} - k x + 8 = 0$ , then one of the values of $k$ is [2000]

1/8
8
4
1/4

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(3)

The directrix of the parabola $y^{2} = 4 a (x - x_{1})$ is given by $x = x_{1} - a .$

Now given parabola is

$y^{2} = k x - 8 \Rightarrow y^{2} = k (x - \frac{8}{k})$

$∴$ $Directrix of parabola is x = \frac{8}{k} - \frac{k}{4};$

Now, $x = 1$ also coincides with $x = \frac{8}{k} - \frac{k}{4}$

On comparing, $\frac{8}{k} - \frac{k}{4} = 1$ $\Rightarrow k^{2} - 4 k - 32 = 0$

$∴$ $k = 4$

Q 11 :

If $x + y = k$ is normal to $y^{2} = 12 x$ , then $k$ is [2000]

$3$
$9$
$- 9$
$- 3$

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(2)

$y = m x + c$ is normal to the parabola $y^{2} = 4 a x$ if $c = - 2 a m - a m^{3}$

Now given line $x + y = k$ is normal to $y^{2} = 12 x$

$∴$ $m = - 1, c = k and a = 3$

$\Rightarrow c = k = - 2 (3) (- 1) - 3 {(- 1)}^{3} = 9$

Q 12 :

A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0, - α)$ to the parabola $x^{2} = - 4 a y$ , where $a > 0$ . Let $L$ be the line passing through $(0, - α)$ and parallel to the directrix of the parabola. Suppose that $L$ intersects the parabola at two points A and B. Let $r$ denote the length of the latus rectum and $s$ denote the square of the length of the line segment AB. If $r : s = 1 : 16$ , then the value of $24 a$ is ________ . [2024]

(12)

$x^{2} = - 4 a y$

$Equation of normal$

$y = m x - 2 a - \frac{a}{m^{2}}$

$- α = - 2 a - \frac{a}{\frac{1}{6}} = - 8 a$

$\Rightarrow α = 8 a$

Equation of required line passing through $(0, - α)$ and parallel to directrix is

$y = - α \Rightarrow y = - 8 a,$ Solving with $x^{2} = - 4 a y$

$x^{2} = 32 a^{2} \Rightarrow x = \pm 4 \sqrt{2} a = \pm \frac{α}{\sqrt{2}}$

$A (\frac{α}{\sqrt{2}}, - α), B (- \frac{α}{\sqrt{2}}, - α) \Rightarrow A B = \sqrt{2} α$

$\Rightarrow \frac{r}{s} = \frac{4 a}{2 α^{2}} = \frac{1}{16} \Rightarrow \frac{4 a}{2 \times 64 a^{2}} = \frac{1}{16}$

$\Rightarrow a = \frac{1}{2} \Rightarrow 24 a = 12$

Q 13 :

Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]

(4)

Let $(t^{2}, 2 t)$ be any point on $y^{2} = 4 x$ and $(h, k)$ be the image of $(t^{2}, 2 t)$ in the line $x + y + 4 = 0 .$

$∴$ $\frac{h - t^{2}}{1} = \frac{k - 2 t}{1} = \frac{- 2 (t^{2} + 2 t + 4)}{2}$

$\Rightarrow h = - (2 t + 4)$ and $k = - (t^{2} + 4)$

For its intersection with $y = - 5$ , we get

$- (t^{2} + 4) = - 5 \Rightarrow t = \pm 1$

$∴$ $A (- 6, - 5) and B (- 2, - 5),$ $∴$ $A B = 4 .$

Q 14 :

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]

(2)

End points of latus rectum of $y^{2} = 4 x$ are $(1, \pm 2)$

Equation of normal to $y^{2} = 4 x$ at (1, 2) is

$y - 2 = - 1 (x - 1) \Rightarrow x + y - 3 = 0$

As it is tangent to circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$

$∴$ $| \frac{3 + (- 2) - 3}{\sqrt{2}} |$ $= r \Rightarrow r^{2} = 2$

Q 15 :

Consider the parabola $y^{2} = 8 x$ . Let $Δ_{1}$ be the area of the triangle formed by the end points of its latus rectum and the point $P (\frac{1}{2}, 2)$ on the parabola and $Δ_{2}$ be the area of the triangle formed by drawing tangents at $P$ and at the end points of the latus rectum. Then $\frac{Δ_{1}}{Δ_{2}}$ is [2011]

(2)

$Δ_{1} = area (∆ P L L') = \frac{1}{2} \times 8 \times \frac{3}{2} = 6$

$Equation of AB is y = 2 x + 1; equation of AC is y = x + 2 and equation of BC is - y = x + 2 .$

On solving the above equations pairwise, we get $A (1, 3), B (- 1, - 1) and C (- 2, 0)$

$∴$ $Δ_{2} = \frac{1}{2}$ $| \begin{matrix} 1 & 3 & 1 \\ - 1 & - 1 & 1 \\ - 2 & 0 & 1 \end{matrix} |$ $= 3$

$∴$ $\frac{Δ_{1}}{Δ_{2}} = 2$

Q 16 :

Let $A_{1}, B_{1}, C_{1}$ be three points in the $x y$ -plane. Suppose that the lines $A_{1} C_{1}$ and $B_{1} C_{1}$ are tangents to the curve $y^{2} = 8 x$ at $A_{1}$ and $B_{1}$ , respectively. If $O = (0, 0)$ and $C_{1} = (- 4, 0)$ , then which of the following statements is (are) TRUE? [2024]

The length of the line segment $O A_{1}$ is $4 \sqrt{3}$
The length of the line segment $A_{1} B_{1}$ is 16
The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(0, 0)$
The orthocenter of the triangle $A_{1} B_{1} C_{1}$ is $(1, 0)$

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Select one or more options

(3, 4)

Let parametric coordinates of $A_{1}$ and $B_{1}$ are

$A_{1} = (2 t_{1}^{2}, 4 t_{1}), B_{1} = (2 t_{2}^{2}, 4 t_{2})$

$C \equiv (- 4, 0) \equiv (2 t_{1} t_{2}, 2 (t_{1} + t_{2}))$

$\Rightarrow t_{2} = - t_{1} and t_{1} (- t_{1}) = - 2$

$t_{1} = \sqrt{2}, t_{2} = - \sqrt{2}$

$A_{1} \equiv (4, 4 \sqrt{2}), B_{1} \equiv (4, - 4 \sqrt{2})$

$∴$ $O A_{1} = \sqrt{4^{2} + {(4 \sqrt{2})}^{2}} = 4 \sqrt{3}$

$Length of line segment A_{1} B_{1} = 8 \sqrt{2}$

$Altitude C_{1} M : y = 0 . . . (i)$

$Altitude B_{1} N : \sqrt{2} x + y = 0 . . . (i i)$

$∴$ $Orthocentre \equiv (0, 0)$

Q 17 :

Consider the parabola $y^{2} = 4 x$ . Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = (- 2, 1)$ meet the parabola at $P_{1}$ and $P_{2} .$ Let $Q_{1}$ and $Q_{2}$ be points on the lines $S P_{1}$ and $S P_{2}$ respectively such that $P Q_{1} ⊥ S P_{1}$ and $P Q_{2} ⊥ S P_{2} .$ Then, which of the following is/are TRUE? [2022]

$S Q_{1} = 2$
$Q_{1} Q_{2} = \frac{3 \sqrt{10}}{5}$
$P Q_{1} = 3$
$S Q_{2} = 1$

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Select one or more options

(1, 2, 3, 4)

Let a rough graph to refer the question

Let parametric point at $P_{1} (t^{2}, 2 t)$ , then tangent at $P_{1}$

$t y = x + t^{2}$

Since it passes through $(- 2, 1)$

$∴$ $t^{2} - t - 2 = 0$

$∴$ $t = 2, - 1$

$∴$ $P_{1} (4, 4) and P_{2} (1, - 2)$

$∴$ $S P_{1} : 4 x - 3 y - 4 = 0 and S P_{2} : x - 1 = 0$

and for $Q_{1} :$ $\frac{x_{1} + 2}{4} = \frac{y_{1} - 1}{- 3} = \frac{- (- 8 - 3 - 4)}{25} = \frac{3}{5}$

$∴$ $x_{1} = \frac{2}{5}, y_{1} = - \frac{4}{5} and Q_{2} = (1, 1)$

$So, S Q_{1} = \sqrt{{(1 - \frac{2}{5})}^{2} + {(\frac{4}{5})}^{2}} = 1$

$Q_{1} Q_{2} = \sqrt{\frac{9}{25} + \frac{81}{25}} = \sqrt{\frac{90}{25}} = \frac{3 \sqrt{10}}{5}$

$P Q_{1} = \sqrt{\frac{144}{25} + \frac{81}{25}} = \frac{15}{5} = 3, S Q_{2} = 1$

Q 18 :

Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $P Q$ and $P Q'$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE? [2021]

The triangle $P F Q$ is a right-angled triangle
The triangle $Q P Q'$ is a right-angled triangle
The distance between P and F is $5 \sqrt{2}$
F lies on the line joining $Q$ and $Q'$

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4

Select one or more options

(1, 2, 4)

Given that

$E : y^{2} = 8 x,$

$P = (- 2, 4)$

Point $P (- 2, 4)$ lies on directrix of parabola.

So, $∠ Q P Q' = \frac{π}{2}$ and chord $Q Q'$ is a focal chord and segment PQ subtends right angle at the focus. So, $∠ P F Q = \frac{π}{2}$

Slope of $Q Q' = \frac{2}{t_{1} + t_{2}} = 1$

Slope of $P F = - 1$ $∴$ $Q Q' ⊥ P F$

$P F = 4 \sqrt{2}$

Q 19 :

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ and midpoint $(h, k)$ , then which of the following is (are) possible value(s) of $p, h$ and $k$ ? [2017]

$p = - 2, h = 2, k = - 4$
$p = - 1, h = 1, k = - 3$
$p = 2, h = 3, k = - 4$
$p = 5, h = 4, k = - 3$

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(3)

If $(h, k)$ is the mid point of chord of parabola $y^{2} = 16 x$ , then equation of chord will be given by

$T = S_{1} \Rightarrow y k - 8 (x + h) = k^{2} - 16 h$

$\Rightarrow 8 x - k y = 8 h - k^{2} ...(i)$

But given, the equation of chord is

$2 x + y = p ...(ii)$

$∴$ $(i) and (ii) are identical lines$

$\Rightarrow \frac{8}{2} = \frac{- k}{1} = \frac{8 h - k^{2}}{p}$

$\Rightarrow k = - 4 and p = 2 h - 4$

which are satisfied by option (3)

Q 20 :

Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0 .$ Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then [2016]

$S P = 2 \sqrt{5}$
$S Q : Q P = (\sqrt{5} + 1) : 2$
the $x$ -intercept of the normal to the parabola at $P$ is 6
the slope of the tangent to the circle at $Q$ is $\frac{1}{2}$

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Select one or more options

(1, 3, 4)

Let point $P$ on parabola $y^{2} = 4 x$ be $(t^{2}, 2 t)$

$∵$ PS is shortest distance, therefore PS should be the normal to parabola.

Equation of normal to $y^{2} = 4 x$ at $P (t^{2}, 2 t)$ is

$y - 2 t = - t (x - t^{2})$

It passes through $S (2, 8)$

$∴$ $8 - 2 t = - t (2 - t^{2})$

$\Rightarrow t^{3} = 8 \Rightarrow t = 2,$ $∴$ $P (4, 4)$

Also slope of tangent to circle at $Q = \frac{- 1}{slope of P S} = \frac{1}{2}$

Equation of normal at $t = 2$ is $2 x + y = 12$

Clearly $x$ -intercept = 6, Now $S P = 2 \sqrt{5}$ and $S Q = r = 2$

$∴$ $Q divides S P in the ratio S P : P Q$

$= 2 : 2 (\sqrt{5} - 1) or (\sqrt{5} + 1) : 4$

Q 21 :

The circle $C_{1} : x^{2} + y^{2} = 3$ , with centre at O, intersects the parabola $x^{2} = 2 y$ at the point P in the first quadrant. Let the tangent to the circle $C_{1}$ , at P touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$ , respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $2 \sqrt{3}$ and centres $Q_{2}$ and $Q_{3}$ , respectively. If $Q_{2}$ and $Q_{3}$ lie on the $y$ -axis, then [2016]

$Q_{2} Q_{3} = 12$
$R_{2} R_{3} = 4 \sqrt{6}$
area of the triangle $O R_{2} R_{3}$ is $6 \sqrt{2}$
area of the triangle $P Q_{2} Q_{3}$ is $4 \sqrt{2}$

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Select one or more options

(1, 2, 3)

Given circle, $C_{1} : x^{2} + y^{2} = 3 ...(i)$

and parabola: $x^{2} = 2 y ...(ii)$

Intersection point of (i) and (ii) in first quadrant

$y^{2} + 2 y - 3 = 0 \Rightarrow y = 1 (∵ y \neq - 3)$

$∴$ $x = \sqrt{2} \Rightarrow P (\sqrt{2}, 1)$

Equation of tangent to circle $C_{1}$ at $P$ is $\sqrt{2} x + y - 3 = 0$

Let centre of circle $C_{2}$ be $(0, k)$ ; $r = 2 \sqrt{3}$

$∴$ $| \frac{k - 3}{\sqrt{3}} |$ $= 2 \sqrt{3} \Rightarrow k = 9 or - 3$

$∴$ $Q_{2} (0, 9), Q_{3} (0, - 3)$

$(1) Q_{2} Q_{3} = 12$

$(2) R_{2} R_{3} = length of transverse common tangent$ $= \sqrt{{(Q_{2} Q_{3})}^{2} - {(r_{1} + r_{2})}^{2}} = \sqrt{{(12)}^{2} - {(4 \sqrt{3})}^{2}} = 4 \sqrt{6}$

$(3) Area (∆ O R_{2} R_{3}) = \frac{1}{2} \times R_{2} R_{3} \times length of ⊥ from origin to tangent$
$= \frac{1}{2} \times 4 \sqrt{6} \times \sqrt{3} = 6 \sqrt{2}$

$(4) ar (∆ P Q_{2} Q_{3}) = \frac{1}{2} \times Q_{2} Q_{3} ⊥ distance of P from y-axis$
$= \frac{1}{2} \times 12 \times \sqrt{2} = 6 \sqrt{2}$

Q 22 :

Let P and Q be distinct points on the parabola $y^{2} = 2 x$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $∆ O P Q$ is $3 \sqrt{2}$ , then which of the following is (are) the coordinates of P? [2015]

$(4, 2 \sqrt{2})$
$(9, 3 \sqrt{2})$
$(\frac{1}{4}, \frac{1}{\sqrt{2}})$
$(1, \sqrt{2})$

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Select one or more options

(1, 4)

Let point P be in first quadrant and lying on parabola $y^{2} = 2 x$ be $(\frac{a^{2}}{2}, a) .$ Let Q be the point $(\frac{b^{2}}{2}, b) .$ Clearly $a > 0 .$

$∵$ $P Q is the diameter of circle through P, O and Q$

$∴$ $∠ P O Q = 90 ° \Rightarrow \frac{a}{a^{2} / 2} \times \frac{b}{b^{2} / 2} = - 1 \Rightarrow a b = - 4$

$\Rightarrow b is negative (∵ a > 0)$ Given area $(∆ P O Q) = 3 \sqrt{2}$

$\Rightarrow \frac{1}{2}$ $| \begin{matrix} 0 & 0 & 1 \\ \frac{a^{2}}{2} & a & 1 \\ \frac{b^{2}}{2} & b & 1 \end{matrix} |$ $= 3 \sqrt{2}$

$\Rightarrow \frac{1}{4} a b (a - b) = \pm 3 \sqrt{2} \Rightarrow a - b = \pm 3 \sqrt{2}$

As $a$ is positive and $b$ is negative, we have $a - b = 3 \sqrt{2}$

$\Rightarrow a + \frac{4}{a} = 3 \sqrt{2} (∵ a b = - 4)$

$\Rightarrow a^{2} - 3 \sqrt{2} a + 4 = 0 \Rightarrow (a - 2 \sqrt{2}) (a - \sqrt{2}) = 0$

$∴$ $a = 2 \sqrt{2}, \sqrt{2}$

$∴$ $Point P can be (\frac{{(2 \sqrt{2})}^{2}}{2}, 2 \sqrt{2}) or (\frac{{(\sqrt{2})}^{2}}{2}, \sqrt{2})$

$i.e. (4, 2 \sqrt{2}) or (1, \sqrt{2})$

Q 23 :

Let L be a normal to the parabola $y^{2} = 4 x$ . If L passes through the point (9, 6), then L is given by [2011]

$y - x + 3 = 0$
$y + 3 x - 33 = 0$
$y + x - 15 = 0$
$y - 2 x + 12 = 0$

1

2

3

4

Select one or more options

(1, 2, 4)

The equation of normal to $y^{2} = 4 x$ is $y = m x - 2 m - m^{3}$

Since the normal passes through (9, 6), $∴$ $6 = 9 m - 2 m - m^{3}$

$\Rightarrow m^{3} - 7 m + 6 = 0 \Rightarrow (m - 1) (m^{2} + m - 6) = 0$

$\Rightarrow (m - 1) (m + 3) (m - 2) = 0 \Rightarrow m = 1, 2, - 3$

$∴$ $Normals are y - x + 3 = 0 or y - 2 x + 12 = 0 or y + 3 x - 33 = 0$

Q 24 :

Let A and B be two distinct points on the parabola $y^{2} = 4 x$ . If the axis of the parabola touches a circle of radius $r$ having AB as its diameter, then the slope of the line joining A and B can be [2010]

$- \frac{1}{r}$
$\frac{1}{r}$
$\frac{2}{r}$
$- \frac{2}{r}$

1

2

3

4

Select one or more options

(3, 4)

Given equation of parabola is $y^{2} = 4 x$

Its axis is $x$ -axis.

Let $A (t_{1}^{2}, 2 t_{1})$ and $B (t_{2}^{2}, 2 t_{2})$

Then centre of circle drawn with AB as diameter is

$(\frac{t_{1}^{2} + t_{2}^{2}}{2}, t_{1} + t_{2})$

As circle touches axis of parabola i.e., $x$ -axis

$∴$ $r =$ $| t_{1} + t_{2} |$ $\Rightarrow t_{1} + t_{2} = \pm r$

$∴$ $Slope of A B = \frac{2 (t_{2} - t_{1})}{t_{2}^{2} - t_{1}^{2}} = \frac{2}{t_{2} + t_{1}} = \pm \frac{2}{r}$

Q 25 :

The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose [2009]

vertex is $(\frac{2 a}{3}, 0)$
directrix is $x = 0$
latus rectum is $\frac{2 a}{3}$
focus is $(a, 0)$

1

2

3

4

Select one or more options

(1, 4)

Let $P (a t^{2}, 2 a t)$ be any point on the parabola $y^{2} = 4 a x .$

$∴$ Tangent to the parabola at P is $y = \frac{x}{t} + a t,$

which meets the axis of parabola i.e. $x$ -axis at $T (- a t^{2}, 0) .$

Also normal to parabola at P is $t x + y = 2 a t + a t^{3}$

which meets the axis of parabola at $N (2 a + a t^{2}, 0) .$

Let $G (x, y)$ be the centroid of $∆ P T N$ , then

$x = \frac{a t^{2} - a t^{2} + 2 a + a t^{2}}{3}$ and $y = \frac{2 a t}{3}$

$\Rightarrow x = \frac{2 a + a t^{2}}{3} ...(i)$ and $y = \frac{2 a t}{3} ...(ii)$

Eliminating $t$ from (i) and (ii), we get the locus of centroid $G$ as

$3 x = 2 a + a {(\frac{3 y}{2 a})}^{2}$ $\Rightarrow y^{2} = \frac{4 a}{3} (x - \frac{2 a}{3}),$

which is a parabola with vertex $(\frac{2 a}{3}, 0),$ directrix as $x - \frac{2 a}{3} = - \frac{a}{3} \Rightarrow x = \frac{a}{3},$ latus rectum as $\frac{4 a}{3}$ and focus as $(a, 0) .$

Q 26 :

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - {(x - 2)}^{2}$ is/are [2006]

$y = 4 (x - 1)$
$y = 0$
$y = - 4 (x - 1)$
$y = - 30 x - 50$

1

2

3

4

Select one or more options

(1, 2)

If $y = m x + c$ is tangent to $y = x^{2}$ then $x^{2} - m x - c = 0 has equal roots$

$\Rightarrow m^{2} + 4 c = 0 \Rightarrow c = - \frac{m^{2}}{4}$

$∴$ $y = m x - \frac{m^{2}}{4} is tangent to y = x^{2}$

$∴$ $This is also tangent to y = - {(x - 2)}^{2}$

$\Rightarrow m x - \frac{m^{2}}{4} = - x^{2} + 4 x - 4$

$\Rightarrow x^{2} + (m - 4) x + (4 - \frac{m^{2}}{4}) = 0 has equal roots$

$∴$ $m^{2} - 8 m + 16 = - m^{2} + 16 \Rightarrow m = 0, 4$

$∴$ $y = 0 or y = 4 x - 4 are the tangents.$

Q 27 :

Match the following: (3, 0) is the point from which three normals are drawn to the parabola $y^{2} = 4 x$ which meet the parabola in the points P, Q and R. Then [2006]

Column I Column II

(A) Area of $∆ P Q R$ (p) 2

(B) Radius of circumcircle of $∆ P Q R$ (q) $\frac{5}{2}$

(C) Centroid of $∆ P Q R$ (r) $(\frac{5}{2}, 0)$

(D) Circumcentre of $∆ P Q R$ (s) $(\frac{2}{3}, 0)$

$(A) \to (q); (B) \to (p); (C) \to (r); (D) \to (s)$
$(A) \to (r); (B) \to (s); (C) \to (q); (D) \to (p)$
$(A) \to (r); (B) \to (q); (C) \to (s); (D) \to (p)$
$(A) \to (p); (B) \to (q); (C) \to (s); (D) \to (r)$

1

2

3

4

(4)

$Let y = m x - 2 m - m^{3} be the equation of normal to y^{2} = 4 x .$

$Since it passes through (3, 0),$ $∴$ $m = 0, 1, - 1$

$Hence three points on parabola are given by (m^{2}, - 2 m) for m = 0, 1, - 1$

$∴$ $P (0, 0), Q (1, 2) and R (1, - 2)$

$∴$ $Area (∆ P Q R) = \frac{1}{2}$ $| \begin{matrix} 0 & 0 & 1 \\ 1 & 2 & 1 \\ 1 & - 2 & 1 \end{matrix} |$ $= 2$

$Radius of circumcircle, R = \frac{a b c}{4 Δ} = \frac{\sqrt{5} \times \sqrt{5} \times 4}{4 \times 2} = \frac{5}{2}$

$(where, a, b, c are the sides of ∆ P Q R)$

$Centroid of ∆ P Q R = (\frac{2}{3}, 0); Circumcentre = (\frac{5}{2}, 0)$

Q 28 :

Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x .$ Suppose that PQ is the focal chord and lines QR and PK are parallel, where $K$ is the point $(2 a, 0)$ . [2014]

Q. The value of $r$ is

$- \frac{1}{t}$
$\frac{t^{2} + 1}{t}$
$\frac{1}{t}$
$\frac{t^{2} - 1}{t}$

1

2

3

4

(4)

$∵$ $P Q is a focal chord, Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$

$Also Q R ∥ P K,$ $∴$ $m_{Q R} = m_{P K}$

$\Rightarrow \frac{\frac{- 2 a}{t} - 2 a r}{\frac{a}{t^{2}} - a r^{2}} = \frac{0 - 2 a t}{2 a - a t^{2}}$

$\Rightarrow \frac{- 2 a (\frac{1}{t} + r)}{a (\frac{1}{t} + r) (\frac{1}{t} - r)} = \frac{- 2 a t}{a (2 - t^{2})}$

$\Rightarrow 2 - t^{2} = t (\frac{1}{t} - r) [∵ r \neq - \frac{1}{t} otherwise Q will coincide with R]$

$\Rightarrow 2 - t^{2} = 1 - t r \Rightarrow r = \frac{t^{2} - 1}{t}$

Q 29 :

Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is the focal chord and lines QR and PK are parallel, where $K$ is the point $(2 a, 0)$ . [2014]

Q. If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

$\frac{{(t^{2} + 1)}^{2}}{2 t^{3}}$
$\frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$
$\frac{a {(t^{2} + 1)}^{2}}{t^{3}}$
$\frac{a {(t^{2} + 2)}^{2}}{t^{3}}$

1

2

3

4

(2)

$Tangent at P is t y = x + a t^{2} . . . (i)$

$Normal at S is s x + y = 2 a s + a s^{3} . . . (i i)$

$Given s t = 1 \Rightarrow s = \frac{1}{t}$

$∴$ $\frac{x}{t} + y = \frac{2 a}{t} + \frac{a}{t^{3}} \Rightarrow x t^{2} + y t^{3} = 2 a t^{2} + a$

Now putting the value of $x$ from equation (i) in above equation, we get

$\Rightarrow t^{2} (t y - a t^{2}) + y t^{3} = 2 a t^{2} + a$

$\Rightarrow 2 t^{3} y = a (t^{4} + 2 t^{2} + 1)$

$∴$ $y = \frac{a (t^{4} + 2 t^{2} + 1)}{2 t^{3}} = \frac{a {(t^{2} + 1)}^{2}}{2 t^{3}}$

Q 30 :

Let $P Q$ be a focal chord of the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a, a > 0$ . [2013]

Q. Length of chord PQ is

$7 a$
$5 a$
$2 a$
$3 a$

1

2

3

4

(2)

$P Q = \sqrt{{(a t^{2} - \frac{a}{t^{2}})}^{2} + {(2 a t + \frac{2 a}{t})}^{2}}$

$= a \sqrt{{(t + \frac{1}{t})}^{2} {(t - \frac{1}{t})}^{2} + 4 {(t + \frac{1}{t})}^{2}}$

$= a (t + \frac{1}{t}) \sqrt{{(t - \frac{1}{t})}^{2} + 4}$

$= a {(t + \frac{1}{t})}^{2} = 5 a$

Topic Question Set

Q 2 :

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1 : 3. Then the locus of $P$ is [2011]

Q 3 :

The axis of a parabola is along the line $y = x$ and the distances of its vertex and focus from origin are $\sqrt{2}$ and $2 \sqrt{2}$ respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is [2006]

Q 4 :

Tangent to the curve $y = x^{2} + 6$ at a point (1, 7) touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ at a point $Q$ . Then the coordinates of $Q$ are [2005]

Q 5 :

The angle between the tangents drawn from the point (1, 4) to the parabola $y^{2} = 4 x$ is [2004]

Q 6 :

The focal chord to $y^{2} = 16 x$ is tangent to ${(x - 6)}^{2} + y^{2} = 2$ , then the possible values of the slope of this chord are [2003]

Q 7 :

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix [2002]

Q 8 :

The equation of the directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is [2001]

Q 9 :

The equation of the common tangent touching the circle ${(x - 3)}^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the $x$ -axis is [2001]

Q 10 :

If the line $x - 1 = 0$ is the directrix of the parabola $y^{2} - k x + 8 = 0$ , then one of the values of $k$ is [2000]

Q 11 :

If $x + y = k$ is normal to $y^{2} = 12 x$ , then $k$ is [2000]

Q 13 :

Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]

Q 14 :

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]

Q 18 :

Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $P Q$ and $P Q'$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE? [2021]

Q 19 :

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ and midpoint $(h, k)$ , then which of the following is (are) possible value(s) of $p, h$ and $k$ ? [2017]

Q 20 :

Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0 .$ Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then [2016]

Q 23 :

Let L be a normal to the parabola $y^{2} = 4 x$ . If L passes through the point (9, 6), then L is given by [2011]

Q 24 :

Let A and B be two distinct points on the parabola $y^{2} = 4 x$ . If the axis of the parabola touches a circle of radius $r$ having AB as its diameter, then the slope of the line joining A and B can be [2010]

Q 25 :

The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose [2009]

Q 26 :

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - {(x - 2)}^{2}$ is/are [2006]

Q 30 :

Let $P Q$ be a focal chord of the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a, a > 0$ . [2013]

Q. Length of chord PQ is

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	Column I		Column II
(A)	Area of $∆ P Q R$	(p)	2
(B)	Radius of circumcircle of $∆ P Q R$	(q)	$\frac{5}{2}$
(C)	Centroid of $∆ P Q R$	(r)	$(\frac{5}{2}, 0)$
(D)	Circumcentre of $∆ P Q R$	(s)	$(\frac{2}{3}, 0)$

Topic Question Set

Q 1 : Let P be a point on the parabola y2=4ax, where a>0. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a,m) is [2023]

Q 2 : Let (x,y) be any point on the parabola y2=4x. Let P be the point that divides the line segment from (0, 0) to (x,y) in the ratio 1 : 3. Then the locus of P is [2011]

Q 3 : The axis of a parabola is along the line y=x and the distances of its vertex and focus from origin are 2 and 22 respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is [2006]

Q 4 : Tangent to the curve y=x2+6 at a point (1, 7) touches the circle x2+y2+16x+12y+c=0 at a point Q. Then the coordinates of Q are [2005]

Q 5 : The angle between the tangents drawn from the point (1, 4) to the parabola y2=4x is [2004]

Q 6 : The focal chord to y2=16x is tangent to (x-6)2+y2=2, then the possible values of the slope of this chord are [2003]

Q 7 : The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2=4ax is another parabola with directrix [2002]

Q 8 : The equation of the directrix of the parabola y2+4y+4x+2=0 is [2001]

Q 9 : The equation of the common tangent touching the circle (x-3)2+y2=9 and the parabola y2=4x above the x-axis is [2001]

Q 10 : If the line x-1=0 is the directrix of the parabola y2-kx+8=0, then one of the values of k is [2000]

Q 11 : If x+y=k is normal to y2=12x, then k is [2000]

Q 13 : Let the curve C be the mirror image of the parabola y2=4x with respect to the line x+y+4=0. If A and B are the points of intersection of C with the line y=-5, then the distance between A and B is [2015]

Q 14 : If the normals of the parabola y2=4x drawn at the end points of its latus rectum are tangents to the circle (x-3)2+(y+2)2=r2, then the value of r2 is [2015]

Q 15 : Consider the parabola y2=8x. Let Δ1 be the area of the triangle formed by the end points of its latus rectum and the point P(12,2) on the parabola and Δ2 be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum. Then Δ1Δ2 is [2011]

Q 16 : Let A1,B1,C1 be three points in the xy-plane. Suppose that the lines A1C1 and B1C1 are tangents to the curve y2=8x at A1 and B1, respectively. If O=(0,0) and C1=(-4,0), then which of the following statements is (are) TRUE? [2024]

Q 18 : Let E denote the parabola y2=8x. Let P=(-2,4) and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE? [2021]

Q 19 : If a chord, which is not a tangent, of the parabola y2=16x has the equation 2x+y=p and midpoint (h,k), then which of the following is (are) possible value(s) of p,h and k? [2017]

Q 20 : Let P be the point on the parabola y2=4x which is at the shortest distance from the center S of the circle x2+y2-4x-16y+64=0. Let Q be the point on the circle dividing the line segment SP internally. Then [2016]

Q 22 : Let P and Q be distinct points on the parabola y2=2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle ∆OPQ is 32, then which of the following is (are) the coordinates of P? [2015]

Q 23 : Let L be a normal to the parabola y2=4x. If L passes through the point (9, 6), then L is given by [2011]

Q 24 : Let A and B be two distinct points on the parabola y2=4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be [2010]

Q 25 : The tangent PT and the normal PN to the parabola y2=4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose [2009]

Q 26 : The equations of the common tangents to the parabola y=x2 and y=-(x-2)2 is/are [2006]

Q 28 : Let a,r,s,t be nonzero real numbers. Let P(at2,2at), Q, R(ar2,2ar) and S(as2,2as) be distinct points on the parabola y2=4ax. Suppose that PQ is the focal chord and lines QR and PK are parallel, where K is the point (2a,0). [2014] Q. The value of r is

Q 30 : Let PQ be a focal chord of the parabola y2=4ax. The tangents to the parabola at P and Q meet at a point lying on the line y=2x+a, a>0. [2013] Q. Length of chord PQ is

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Q 2 :

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from (0, 0) to $(x, y)$ in the ratio 1 : 3. Then the locus of $P$ is [2011]

Q 3 :

The axis of a parabola is along the line $y = x$ and the distances of its vertex and focus from origin are $\sqrt{2}$ and $2 \sqrt{2}$ respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is [2006]

Q 4 :

Tangent to the curve $y = x^{2} + 6$ at a point (1, 7) touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ at a point $Q$ . Then the coordinates of $Q$ are [2005]

Q 5 :

The angle between the tangents drawn from the point (1, 4) to the parabola $y^{2} = 4 x$ is [2004]

Q 6 :

The focal chord to $y^{2} = 16 x$ is tangent to ${(x - 6)}^{2} + y^{2} = 2$ , then the possible values of the slope of this chord are [2003]

Q 7 :

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2} = 4 a x$ is another parabola with directrix [2002]

Q 8 :

The equation of the directrix of the parabola $y^{2} + 4 y + 4 x + 2 = 0$ is [2001]

Q 9 :

The equation of the common tangent touching the circle ${(x - 3)}^{2} + y^{2} = 9$ and the parabola $y^{2} = 4 x$ above the $x$ -axis is [2001]

Q 10 :

If the line $x - 1 = 0$ is the directrix of the parabola $y^{2} - k x + 8 = 0$ , then one of the values of $k$ is [2000]

Q 11 :

If $x + y = k$ is normal to $y^{2} = 12 x$ , then $k$ is [2000]

Q 13 :

Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0 .$ If A and B are the points of intersection of $C$ with the line $y = - 5$ , then the distance between A and B is [2015]

Q 14 :

If the normals of the parabola $y^{2} = 4 x$ drawn at the end points of its latus rectum are tangents to the circle ${(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}$ , then the value of $r^{2}$ is [2015]

Q 18 :

Let $E$ denote the parabola $y^{2} = 8 x$ . Let $P = (- 2, 4)$ and let $Q$ and $Q'$ be two distinct points on $E$ such that the lines $P Q$ and $P Q'$ are tangents to $E$ . Let $F$ be the focus of $E$ . Then which of the following statements is (are) TRUE? [2021]

Q 19 :

If a chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation $2 x + y = p$ and midpoint $(h, k)$ , then which of the following is (are) possible value(s) of $p, h$ and $k$ ? [2017]

Q 20 :

Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0 .$ Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then [2016]

Q 23 :

Let L be a normal to the parabola $y^{2} = 4 x$ . If L passes through the point (9, 6), then L is given by [2011]

Q 24 :

Let A and B be two distinct points on the parabola $y^{2} = 4 x$ . If the axis of the parabola touches a circle of radius $r$ having AB as its diameter, then the slope of the line joining A and B can be [2010]

Q 25 :

The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose [2009]

Q 26 :

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - {(x - 2)}^{2}$ is/are [2006]

Q 30 :

Let $P Q$ be a focal chord of the parabola $y^{2} = 4 a x$ . The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a, a > 0$ . [2013]

Q. Length of chord PQ is