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

Q 11 :

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. If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $\tan θ =$

$\frac{2}{3} \sqrt{7}$
$\frac{- 2}{3} \sqrt{7}$
$\frac{2}{3} \sqrt{5}$
$\frac{- 2}{3} \sqrt{5}$

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(4)

$Since P Q is the focal chord of y^{2} = 4 a x$

$∴$ $Coordinates of P and Q can be taken as$

$P (a t^{2}, 2 a t) and Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$

$Equation of tangents at P and Q are$

$y = \frac{x}{t} + a t and y = - x t - \frac{a}{t},$

$which intersect each other at R (- a, a (t - \frac{1}{t}))$

$As R lies on the y = 2 x + a, a > 0$

$∴$ $a (t - \frac{1}{t}) = - 2 a + a \Rightarrow t - \frac{1}{t} = - 1 \Rightarrow t + \frac{1}{t} = \sqrt{5}$

$Now, m_{O P} = \frac{2}{t} and m_{O Q} = - 2 t$

$∴$ $\tan θ = \frac{\frac{2}{t} + 2 t}{1 - 4} = \frac{2 (t + \frac{1}{t})}{- 3} = \frac{2 \sqrt{5}}{- 3}$

Q 12 :

Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$ -axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$ -axis at $S$ . [2007]

Q. The ratio of the areas of the triangles $P Q S$ and $P Q R$ is

$1 : \sqrt{2}$
$1 : 2$
$1 : 4$
$1 : 8$

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(3)

$\frac{area (∆ P Q S)}{area (∆ P Q R)} = \frac{\frac{1}{2} P Q \times S T}{\frac{1}{2} P Q \times T R} = \frac{S T}{T R} = \frac{2}{8} = \frac{1}{4}$

Q 13 :

Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the $x$ -axis at R and tangents to the parabola at P and Q intersect the $x$ -axis at S. [2007]

Q. The radius of the circumcircle of the triangle $P R S$ is

$5$
$3 \sqrt{3}$
$3 \sqrt{2}$
$2 \sqrt{3}$

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(2)

$For ∆ P R S,$

$area (∆ P R S) = Δ = \frac{1}{2} \times S R \times P T = \frac{1}{2} \times 10 \times 2 \sqrt{2} = 10 \sqrt{2},$

$a = P S = 2 \sqrt{3}, b = P R = 6 \sqrt{2}, c = S R = 10$

$∴$ $Radius of circumcircle of ∆ P R S,$

$R = \frac{a b c}{4 Δ} = \frac{2 \sqrt{3} \times 6 \sqrt{2} \times 10}{4 \times 10 \sqrt{2}} = 3 \sqrt{3}$

Q 14 :

Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the $x$ -axis at R and tangents to the parabola at P and Q intersect the $x$ -axis at S. [2007]

Q. The radius of the incircle of the triangle PQR is

$4$
$3$
$\frac{8}{3}$
$2$

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(4)

$Radius of incircle$

$r = \frac{Δ}{s} = \frac{area (∆ P Q R)}{semi-perimeter of ∆ P Q R}$

$Here a = P R = 6 \sqrt{2}, b = Q P = 6 \sqrt{2}, c = P Q = 4 \sqrt{2}$

$and area (∆ P Q R) = \frac{1}{2} \times P Q \times T R = 16 \sqrt{2}$

$∴$ $Perimeter of ∆ P Q R = \frac{6 \sqrt{2} + 6 \sqrt{2} + 4 \sqrt{2}}{2} = 8 \sqrt{2}$

$∴$ $r = \frac{16 \sqrt{2}}{8 \sqrt{2}} = 2$

Q 15 :

STATEMENT-1: The curve $y = \frac{- x^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ , because

STATEMENT-2: A parabola is symmetric about its axis. [2007]

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
Statement-1 is True, Statement-2 is False
Statement-1 is False, Statement-2 is True

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(1)

$Given curve is y = - \frac{x^{2}}{2} + x + 1$

$\Rightarrow {(x - 1)}^{2} = - 2 (y - \frac{3}{2}), which is a parabola.$

$It is symmetric with respect to its axis x - 1 = 0$

$∴$ $Both the statements are true and statement-2 is a correct explanation for statement-1.$

Topic Question Set

Q 11 :

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. If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $\tan θ =$

Q 15 :

STATEMENT-1: The curve $y = \frac{- x^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ , because

STATEMENT-2: A parabola is symmetric about its axis. [2007]

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Topic Question Set

Q 11 : 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. If chord PQ subtends an angle θ at the vertex of y2=4ax, then tanθ=

Q 15 : STATEMENT-1: The curve y=-x22+x+1 is symmetric with respect to the line x=1, because STATEMENT-2: A parabola is symmetric about its axis. [2007]

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Q 11 :

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. If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then $\tan θ =$

Q 15 :

STATEMENT-1: The curve $y = \frac{- x^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ , because

STATEMENT-2: A parabola is symmetric about its axis. [2007]