Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 21 :

Let the parabola $y = x^{2} + p x + q$ passing through the point (1, -1) be such that the distance between its vertex and the $x$ -axis is minimum. Then the value of $p^{2} + q^{2}$ is : [2026]

2
4
5
8

1

2

3

4

(2)

We have, $y = x^{2} + p x + q$

It passes through (1, -1)

$∴$ $- 1 = 1 + p + q$

$\Rightarrow p + q = - 2$ $\dots (i)$

$Vertex = (\frac{- p}{2}, - (\frac{p^{2} - 4 q}{4}))$

$Distance from x -axis = (\frac{p^{2} - 4 q}{4})$

$= \frac{p^{2} - 4 (- 2 - p)}{4}$ $[Using (i)]$

$= \frac{p^{2} + 4 p + 8}{4} = \frac{{(p + 2)}^{2} + 4}{4}$

$For minimum distance p = - 2$

$\Rightarrow q = 0$

$So, p^{2} + q^{2} = 4$

Q 22 :

Let $P$ be a moving point on the circle $x^{2} + y^{2} - 6 x - 8 y + 21 = 0$ . Then, the maximum distance of $P$ from the vertex of the parabola $x^{2} + 6 x + y + 13 = 0$ is equal to: [2026]

8
10
12
9

1

2

3

4

(3)

We have, $x^{2} + y^{2} - 6 x - 8 y + 21 = 0$

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

Centre, C = (3, 4) and radius, $r = 2$

Given, $x^{2} + 6 x + y + 13 = 0$

$\Rightarrow y = - x^{2} - 6 x - 13$

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

$∴$ Vertex, V = (-3, -4)

Distance between centre C(3, 4) and vertex (-3, -4)

$= \sqrt{{(3 + 3)}^{2} + {(4 + 4)}^{2}} = \sqrt{36 + 64} = 10$

$∴$ Maximum distance $= 10 + r = 10 + 2 = 12$ units

Q 23 :

Let the point $P$ be the vertex of the parabola $y = x^{2} - 6 x + 12$ . If a line passing through the point $P$ intersects the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ at the points $R$ and $S$ , then the maximum value of ${(P R + P S)}^{2}$ is : [2026]

10
20
25
5

1

2

3

4

(2)

Given, $y = x^{2} - 6 x + 12$

$Vertex (3, 3) so, P (3, 3)$

$Circle : x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ $[Given]$

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

$Centre, C \equiv (1, 2), radius, r = \sqrt{2}$

$So, P C = \sqrt{5}$

$∴$ $P C = \sqrt{5} > \sqrt{2}, point P lies outside$ $the circle.$

$Now, P R = P C - r = \sqrt{5} - \sqrt{2}$

$P S = P C + r = \sqrt{5} + \sqrt{2}$

$P R + P S = 2 \sqrt{5}$

$∴$ ${(P R + P S)}^{2} = {(2 \sqrt{5})}^{2} = 20$

Q 24 :

Let the directrix of the parabola $P : y^{2} = 8 x$ , cut $x$ -axis at the point $A$ . Let $B (α, β)$ , $α > 1$ , be a point on $P$ such that the slope of $A B$ is 3/5. If $B C$ is a focal chord of $P$ , then six times the area of $Δ A B C$ is : [2026]

80
160
174
192

1

2

3

4

(2)

Given, parabola is $y^{2} = 8 x$

Compare with $y^{2} = 4 a x$ , we get $a = 2$

$∴$ Focus (2, 0) and directrix $x = - 2$

$∴$ A(-2, 0)

Point B lies on parabola, so, $β^{2} = 8 α$

Slope of $A B = \frac{3}{5}$ [Given]

$∴$ $\frac{β - 0}{α + 2} = \frac{3}{5} \Rightarrow β = \frac{3}{5} (α + 2)$

$\Rightarrow \frac{9}{25} {(α + 2)}^{2} = 8 α$

$\Rightarrow 9 {(α + 2)}^{2} = 200 α$

$\Rightarrow 9 α^{2} - 164 α + 36 = 0$

$\Rightarrow α = 18$ $(∵ α > 1)$

$∴$ $β^{2} = 144$

$\Rightarrow β = 12$

$∴$ $B (18, 12)$

Now focal chord BC, then

Slope $= \frac{12 - 0}{18 - 2} = \frac{12}{16} = \frac{3}{4}$

$∴$ Equation of $B C : y = \frac{3}{4} x - \frac{3}{2}$ it intersects with parabola then $C (\frac{2}{9}, \frac{- 4}{3})$

Area of $Δ A B C = \frac{1}{2}$ $| \begin{matrix} - 2 & 0 & 1 \\ 18 & 12 & 1 \\ 2 / 9 & - 4 / 3 & 1 \end{matrix} |$

$= \frac{1}{2}$ $| \frac{- 160}{3} |$ $= \frac{80}{3}$ sq. units

$∴$ $6 \times Area of Δ A B C = 6 \times \frac{80}{3} = 160 sq. units$

Q 25 :

Let one root of the quadratic equation in $x$ :

$(k^{2} - 15 k + 27) x^{2} + 9 (k - 1) x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^{2} = 6 k x$ is equal to : [2026]

4
6
8
12

1

2

3

4

(4)

Let $α$ and $2 α$ be the roots of the equation,

$(k^{2} - 15 k + 27) x^{2} + 9 (k - 1) x + 18 = 0$

Sum of roots : $α + 2 α = \frac{- 9 (k - 1)}{k^{2} - 15 k + 27}$

$\Rightarrow α = \frac{- 3 (k - 1)}{k^{2} - 15 k + 27}$ $... (i)$

Product of roots : $α (2 α) = \frac{18}{k^{2} - 15 k + 27}$

$\Rightarrow α^{2} = \frac{9}{k^{2} - 15 k + 27}$ $... (ii)$

From (i) and (ii), we have

$\frac{9 {(k - 1)}^{2}}{{(k^{2} - 15 k + 27)}^{2}} = \frac{9}{k^{2} - 15 k + 27}$

$\Rightarrow {(k - 1)}^{2} = k^{2} - 15 k + 27$

$\Rightarrow k^{2} - 2 k + 1 = k^{2} - 15 k + 27$

$\Rightarrow 13 k = 26$

$\Rightarrow k = 2$

Now parabola is $y^{2} = 12 x$

Length of latus rectum $= 4 \times 3 = 12$

Q 26 :

Let chord PQ of length $3 \sqrt{13}$ of the parabola $y^{2} = 12 x$ be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle $α$ at the focus of the parabola, then $\sin α$ is equal to : [2026]

$\frac{3}{5}$
$\frac{4}{5}$
$\frac{5}{13}$
$\frac{12}{13}$

1

2

3

4

(1)

Let the points be $P (3 t^{2}, 6 t)$ , $Q (12 t^{2}, 12 t)$

$∴$ $P Q = \sqrt{81 t^{4} + 36 t^{2}}$

Also, $P Q = 3 \sqrt{13}$ $(∵ Given)$

So, $81 t^{4} + 36 t^{2} = 117$

$\Rightarrow 9 t^{4} + 4 t^{2} - 13 = 0$

$\Rightarrow (t^{2} - 1) (9 t^{2} + 13) = 0$

$\Rightarrow t = 1$

So, points are P(3, 6), Q(12, 12)

$Slope of Q S = \frac{12 - 0}{12 - 3} = \frac{12}{9} = \frac{4}{3}$

So, $\tan α = \frac{3}{4}$ , $\sin α = \frac{3}{5}$

Q 27 :

Let O be the vertex of the parabola $y^{2} = 4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is : [2026]

1
2
4
8

1

2

3

4

(2)

$t_{1} t_{2} = - 4$

$2 h = t_{1}^{2} + t_{2}^{2};$ $2 k = 2 (t_{1} + t_{2})$

$\Rightarrow 2 h = k^{2} - 2 t_{1} t_{2}$

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

$\Rightarrow k^{2} = 2 (h - 4)$ $∴$ $y^{2} = 2 (x - 4)$

$\Rightarrow L R = 2$

Q 28 :

Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola $y^{2} = 16 x$ . Let the vertex B containing the right angle be (4, 8) and the locus of the centroid of $∆ A B C$ be a conic $C_{0}$ . Then three times the length of latus rectum of $C_{0}$ is _______ . [2026]

(16)

$y^{2} = 4 a x$

For point (4, 8)

$64 = 4 a (4)$

$\Rightarrow 16 a = 64 \Rightarrow a = 4$

$m_{A B} = \frac{8 t_{1} - 8}{4 t_{1}^{2} - 4} = \frac{2}{t_{1} + 1}$

$m_{B C} = \frac{2}{t_{2} + 1}$

$m_{A B} \times m_{B C} = - 1$

$\frac{2}{(t_{1} + 1)} \times \frac{2}{(t_{2} + 1)} = - 1$

$4 = - (t_{1} + 1) (t_{2} + 1)$

$\Rightarrow t_{1} t_{2} = - (t_{1} + t_{2} + 5)$

Centroid of $Δ A B C$ is

$G (h, k) \equiv (\frac{4 + 4 t_{1}^{2} + 4 t_{2}^{2}}{3}, \frac{8 + 8 t_{1} + 8 t_{2}}{3})$

$∴$ $h = \frac{4}{3} (t_{1}^{2} + t_{2}^{2} + 1)$ $...(i)$

$and k = \frac{8}{3} (t_{1} + t_{2} + 1)$

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

$\Rightarrow t_{1}^{2} + t_{2}^{2} = {(t_{1} + t_{2})}^{2} + 2 (t_{1} + t_{2} + 5)$

$\Rightarrow (t_{1}^{2} + t_{2}^{2}) = 3 h - 4$ $[Using (i)]$

Now, $k = \frac{8}{3} + \frac{8}{3} (t_{1} + t_{2})$

$\Rightarrow 3 k - 8 = \frac{8}{3} (t_{1} + t_{2})$

Substitution in the value

$k^{2} = \frac{16}{3} (h - \frac{40}{3})$

Length of latus rectum $= \frac{16}{3}$

So, 3 times the length of latus rectum $= 3 \times \frac{16}{3} = 16$

Topic Question Set

Q 21 :

Let the parabola $y = x^{2} + p x + q$ passing through the point (1, -1) be such that the distance between its vertex and the $x$ -axis is minimum. Then the value of $p^{2} + q^{2}$ is : [2026]

Q 22 :

Let $P$ be a moving point on the circle $x^{2} + y^{2} - 6 x - 8 y + 21 = 0$ . Then, the maximum distance of $P$ from the vertex of the parabola $x^{2} + 6 x + y + 13 = 0$ is equal to: [2026]

Q 23 :

Let the point $P$ be the vertex of the parabola $y = x^{2} - 6 x + 12$ . If a line passing through the point $P$ intersects the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ at the points $R$ and $S$ , then the maximum value of ${(P R + P S)}^{2}$ is : [2026]

Q 24 :

Let the directrix of the parabola $P : y^{2} = 8 x$ , cut $x$ -axis at the point $A$ . Let $B (α, β)$ , $α > 1$ , be a point on $P$ such that the slope of $A B$ is 3/5. If $B C$ is a focal chord of $P$ , then six times the area of $Δ A B C$ is : [2026]

Q 25 :

Let one root of the quadratic equation in $x$ :

$(k^{2} - 15 k + 27) x^{2} + 9 (k - 1) x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^{2} = 6 k x$ is equal to : [2026]

Q 26 :

Let chord PQ of length $3 \sqrt{13}$ of the parabola $y^{2} = 12 x$ be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle $α$ at the focus of the parabola, then $\sin α$ is equal to : [2026]

Q 27 :

Let O be the vertex of the parabola $y^{2} = 4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is : [2026]

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Topic Question Set

Q 21 : Let the parabola y=x2+px+q passing through the point (1, -1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2 is : [2026]

Q 22 : Let P be a moving point on the circle x2+y2-6x-8y+21=0. Then, the maximum distance of P from the vertex of the parabola x2+6x+y+13=0 is equal to: [2026]

Q 23 : Let the point P be the vertex of the parabola y=x2-6x+12. If a line passing through the point P intersects the circle x2+y2-2x-4y+3=0 at the points R and S, then the maximum value of (PR+PS)2 is : [2026]

Q 24 : Let the directrix of the parabola P:y2=8x, cut x-axis at the point A. Let B(α,β), α>1, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of ΔABC is : [2026]

Q 25 : Let one root of the quadratic equation in x: (k2-15k+27)x2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kx is equal to : [2026]

Q 26 : Let chord PQ of length 313 of the parabola y2=12x be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle α at the focus of the parabola, then sinα is equal to : [2026]

Q 27 : Let O be the vertex of the parabola y2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is : [2026]

Q 28 : Let A, B and C be the vertices of a variable right angled triangle inscribed in the parabola y2=16x. Let the vertex B containing the right angle be (4, 8) and the locus of the centroid of ∆ABC be a conic C0. Then three times the length of latus rectum of C0 is _______ . [2026]

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Q 21 :

Let the parabola $y = x^{2} + p x + q$ passing through the point (1, -1) be such that the distance between its vertex and the $x$ -axis is minimum. Then the value of $p^{2} + q^{2}$ is : [2026]

Q 22 :

Let $P$ be a moving point on the circle $x^{2} + y^{2} - 6 x - 8 y + 21 = 0$ . Then, the maximum distance of $P$ from the vertex of the parabola $x^{2} + 6 x + y + 13 = 0$ is equal to: [2026]

Q 23 :

Let the point $P$ be the vertex of the parabola $y = x^{2} - 6 x + 12$ . If a line passing through the point $P$ intersects the circle $x^{2} + y^{2} - 2 x - 4 y + 3 = 0$ at the points $R$ and $S$ , then the maximum value of ${(P R + P S)}^{2}$ is : [2026]

Q 24 :

Let the directrix of the parabola $P : y^{2} = 8 x$ , cut $x$ -axis at the point $A$ . Let $B (α, β)$ , $α > 1$ , be a point on $P$ such that the slope of $A B$ is 3/5. If $B C$ is a focal chord of $P$ , then six times the area of $Δ A B C$ is : [2026]

Q 25 :

Let one root of the quadratic equation in $x$ :

$(k^{2} - 15 k + 27) x^{2} + 9 (k - 1) x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^{2} = 6 k x$ is equal to : [2026]

Q 26 :

Let chord PQ of length $3 \sqrt{13}$ of the parabola $y^{2} = 12 x$ be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle $α$ at the focus of the parabola, then $\sin α$ is equal to : [2026]

Q 27 :

Let O be the vertex of the parabola $y^{2} = 4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is : [2026]