Let P(x_1, y_1) and Q(x_2, y_2) be two distinct points on the ellipse (x^2)/(9)+(y^2)/(4)=1 such that y_1>0 , and y_2>0 . Let C denote the circle x^2+y^2=9 , and M be the point (3,0) . Suppose the line x=x_1 intersects

Correct answer: A. The equation of the line joining P and Q is 2 x+3 y=3(1+ 3 ) aligned & P (3 30^ , 2 30^ ) ( 3 3 2 , 1) & Q (3 60^ , 2 60^ ) ((3)/(2), 3 ) & R( 3 3 2 , (3)/(2)), S((3)/(2), 3 3 2 ) aligned Slope of PQ=m_ PQ = 3 -1 (3)/(2)- 3 3 2 =-(2)/(3) Equation of line PQ y- 3 =-(2)/(3)(x-(3)/(2)) 2 x+3 y=3( 3 +1) option (A) is correct Now aligned & if N_2=(x_2, 0)=((3)/(2), 0) & |N_2 Q|= 3 and |N_2 S|= 3 3 2 aligned 3| N_2 Q|=2| N_2 S| option (C) is correct Now, if N_1=(x_1, 0) N_1=( 3 3 2 , 0) |N_1 P|=1,| N_1 R|=(3)/(2) option (D) is incorrect

JEE Advanced 2025Paper 2

Exam: JEE Advanced
Year: 2025
Paper: Paper 2
Subject: Mathematics
Chapter: Ellipse

JEE Advanced 2025 · Paper 2 · Mathematics · Ellipse

Let P(x_1, y_1) and Q(x_2, y_2) be two distinct points on the...

Let

P\left(x_1, y_1\right)

and

Q\left(x_2, y_2\right)

be two distinct points on the ellipse

\frac{x^2}{9}+\frac{y^2}{4}=1

such that

y_1>0

, and

y_2>0

. Let C denote the circle

x^2+y^2=9

, and

M

be the point

(3,0)

. Suppose the line

x=x_1

intersects

C

R

, and the line

x=x_2

intersects

C

S

, such that the

y

-coordinates of

R

and

S

are positive. Let

\angle R O M=\frac{\pi}{6}

and

\angle S O M=\frac{\pi}{3}

, where

O

denotes the origin

(0,0)

. Let

|X Y|

denote the length of the line segment

X Y

.
Then which of the following statements is (are) TRUE?

Options

AThe equation of the line joining $P$ and $Q$ is $2 x+3 y=3(1+\sqrt{3})$
BThe equation of the line joining $P$ and $Q$ is $2 x+y=3(1+\sqrt{3})$
CIf $N_2=\left(x_2, 0\right)$ , then $3\left|N_2 Q\right|=2\left|N_2 S\right|$
DIf $N_1=\left(x_1, 0\right)$ , then $9\left|N_1 P\right|=4\left|N_1 R\right|$

Correct answer

A. The equation of the line joining $P$ and $Q$ is $2 x+3 y=3(1+\sqrt{3})$

Step-by-step solution

\begin{aligned} & \mathrm{P} \equiv\left(3 \cos 30^{\circ}, 2 \sin 30^{\circ}\right) \equiv\left(\frac{3 \sqrt{3}}{2}, 1\right) \\ & \mathrm{Q} \equiv\left(3 \cos 60^{\circ}, 2 \sin 60^{\circ}\right) \equiv\left(\frac{3}{2}, \sqrt{3}\right) \\ & \mathrm{R}\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right), \mathrm{S}\left(\frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)\end{aligned}

Slope of

\mathrm{PQ}=\mathrm{m}_{\mathrm{PQ}}=\frac{\sqrt{3}-1}{\frac{3}{2}-\frac{3 \sqrt{3}}{2}}=-\frac{2}{3}

Equation of line PQ

\mathrm{y}-\sqrt{3}=-\frac{2}{3}\left(\mathrm{x}-\frac{3}{2}\right) \Rightarrow 2 x+3 y=3(\sqrt{3}+1) \quad

option (A) is correct Now

\begin{aligned} & \text{ if } \mathrm{N}_2=\left(\mathrm{x}_2, 0\right)=\left(\frac{3}{2}, 0\right) \\ & \left|\mathrm{N}_2 \mathrm{Q}\right|=\sqrt{3} \text{\text{ and } }\left|\mathrm{N}_2 \mathrm{ S}\right|=\frac{3 \sqrt{3}}{2}\end{aligned}

\Rightarrow 3\left|\mathrm{ N}_2 \mathrm{Q}\right|=2\left|\mathrm{ N}_2 \mathrm{ S}\right| \quad

option (C) is correct Now, if

\mathrm{N}_1=\left(\mathrm{x}_1, 0\right) \Rightarrow \mathrm{N}_1=\left(\frac{3 \sqrt{3}}{2}, 0\right) \Rightarrow\left|\mathrm{N}_1 \mathrm{P}\right|=1,\left|\mathrm{ N}_1 \mathrm{R}\right|=\frac{3}{2} \quad$ option (D) is incorrect

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