Let and be two distinct points on the ellipse such that and . Let denote the circle and be the point (3, 0). Suppose the line intersects at , and the line intersects at , such that the -coordinates of and are positive. Let and where denote

Question

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ 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 $C$ at $R$ , and the line $x = x_{2}$ intersects $C$ at $S$ , such that the $y$ -coordinates of $R$ and $S$ are positive. Let $∠ R O M = \frac{π}{6}$ and $∠ S O M = \frac{π}{3},$ where $O$ denotes the origin (0, 0). Let $| X Y |$ denote the length of the line segment $X Y$ .

Then which of the following statement(s) is (are) TRUE [2025]

Then which of the following statement(s) is (are) TRUE? [2025]

Smart Achievers · Accepted Answer

(1, 3)

$m_{P Q} = \frac{\sqrt{3} - 1}{\frac{3}{2} (1 - \sqrt{3})} = \frac{- 2}{3}$

$So, equation of P Q is 2 x + 3 y = 3 + 3 \sqrt{3}$

$\Rightarrow 2 x + 3 y = 3 (1 + \sqrt{3})$

$3 | N_{2} Q | = 2 | N_{2} S | \Rightarrow 3 \times \sqrt{3} = 2 \times \frac{3 \sqrt{3}}{2}$

$9 | N_{1} P | = 4 | N_{1} R | \Rightarrow 9 \times 1 = 4 \times \frac{3}{2}$

Solution

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

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

3 If $N_{2} = (x_{2}, 0)$ , then $3 | N_{2} Q | = 2 | N_{2} S |$

4 If $N_{1} = (x_{1}, 0)$ , then $9 | N_{1} P | = 4 | N_{1} R |$

Want Us to Email you About Special Offers & Updates?

Solution

1 The equation of the line joining P and Q is 2x+3y=3(1+3)

2 The equation of the line joining P and Q is 2x+y=3(1+3)

3 If N2=(x2,0), then 3|N2Q|=2|N2S|

4 If N1=(x1,0), then 9|N1P|=4|N1R|

Ans. (1, 3) mPQ=3-132(1-3)=-23 So, equation of PQ is 2x+3y=3+33 ⇒2x+3y=3(1+3) 3|N2Q|=2|N2S|⇒3×3=2×332 9|N1P|=4|N1R|⇒9×1=4×32

Want Us to Email you About Special Offers & Updates?

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

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

3 If $N_{2} = (x_{2}, 0)$ , then $3 | N_{2} Q | = 2 | N_{2} S |$

4 If $N_{1} = (x_{1}, 0)$ , then $9 | N_{1} P | = 4 | N_{1} R |$