Question

Consider an ellipse with major & minor axis having length 4 & $2\sqrt{3}$ respectively. Let P be any variable point on it. S & S' be the foci and C be the centre of ellipse Match List-I with List-Il and select the correct answer using the code given below the list.

	List-I		List-II
(P)	Maximum value of PS.PS' is	(1)	8
(Q)	Minimum value of $(PS)^2 + (PS')^2$ is	(2)	$2+\sqrt{3}$
(R)	Minimum length of portion of tangent at P intercepted between principal axes is	(3)	4
(S)	If foot of normal from P on major axis is P' and normal at P cuts major axis at N, then $\frac{CN}{CP'}$ is	(4)	$\frac{1}{4}$
		(5)	5

• A. P-3, Q-1, R-2, S-4
• B. P-1, Q-3, R-2, S-4
• C. P-3, Q-1, R-4, S-2
• D. P-1, Q-3, R-4, S-2

Answer 1

Answer: (A)

P-3, Q-1, R-2, S-4

Answer 2

1. Ellipse Parameters: Given major axis length $2a = 4 \implies a = 2$. Given minor axis length $2b = 2\sqrt{3} \implies b = \sqrt{3}$. The equation of the ellipse is $\frac{x^2}{4} + \frac{y^2}{3} = 1$. The eccentricity $e$ is found using $b^2 = a^2(1-e^2) \implies 3 = 4(1-e^2) \implies e^2 = 1 - \frac{3}{4} = \frac{1}{4} \implies e = \frac{1}{2}$. The foci S and S' are at $(\pm ae, 0) = (\pm 1, 0)$. The center C is at $(0,0)$.

2. (P) Maximum value of PS.PS': For any point P on the ellipse, $PS + PS' = 2a = 4$. The product $PS \cdot PS'$ is maximized when $PS = PS'$, which occurs at the endpoints of the minor axis. In this case, $PS = PS' = a = 2$. The maximum value of $PS \cdot PS'$ is $a^2 = 2^2 = 4$. This matches List-II (3).

3. (Q) Minimum value of $(PS)^2 + (PS')^2$: Let $PS = r_1$ and $PS' = r_2$. We have $r_1 + r_2 = 2a = 4$. We want to minimize $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = (2a)^2 - 2r_1 r_2 = 4a^2 - 2r_1 r_2$. To minimize this expression, we need to maximize the product $r_1 r_2$. As shown in (P), the maximum value of $r_1 r_2$ is $a^2$. Therefore, the minimum value of $(PS)^2 + (PS')^2$ is $4a^2 - 2a^2 = 2a^2 = 2(2^2) = 8$. This matches List-II (1).

4. (R) Minimum length of portion of tangent at P intercepted between principal axes: The equation of the tangent at P$(a\cos\theta, b\sin\theta)$ is $\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1$. The x-intercept is $X = a/\cos\theta$ and the y-intercept is $Y = b/\sin\theta$. The length of the intercepted portion is $L = \sqrt{X^2 + Y^2} = \sqrt{\frac{a^2}{\cos^2\theta} + \frac{b^2}{\sin^2\theta}}$. The minimum value of $L$ is $a+b$. Substituting $a=2$ and $b=\sqrt{3}$, the minimum length is $2+\sqrt{3}$. This matches List-II (2).

5. (S) $\frac{CN}{CP'}$: Let P be $(a\cos\theta, b\sin\theta)$. The normal at P intersects the major axis (x-axis) at N. The x-coordinate of N is $x_N = ae^2\cos\theta$. The center C is at $(0,0)$, so $CN = |ae^2\cos\theta|$. P' is the foot of the perpendicular from P to the major axis, so P' is $(a\cos\theta, 0)$. The distance $CP' = |a\cos\theta|$. Therefore, the ratio $\frac{CN}{CP'} = \frac{|ae^2\cos\theta|}{|a\cos\theta|} = e^2$. Since $e = 1/2$, $e^2 = 1/4$. This matches List-II (4).

The correct matching is (P)-(3), (Q)-(1), (R)-(2), (S)-(4).