(x^2/(25)+(y^2/(16)=1. List-I (P) The product of the perpendicular distances from the two foci of E to a variable tangent of E. (Q) The radius of the…

The foci are $(\pm3,0)$ since $c^{2}=25-16=9$. (P) For the tangent $\frac{x\cos\theta}{5}+\frac{y\sin\theta}{4}=1$ the two focal distances multiply to $b^{2}=16$, which is (2). (Q) The foot of the perpendicular from a focus to any tangent lies on the auxiliary circle $x^{2}+y^{2}=a^{2}$, of radius $5$, which is (3). (R) The director circle is $x^{2}+y^{2}=a^{2}+b^{2}=41$, so the square of its radius is $41$, which is (4). (S) The tangent has intercepts $\frac5{\cos\theta}$ and $\frac4{\sin\theta}$, so the triangle area is $\frac{10}{\sin\theta\cos\theta}=\frac{20}{\sin2\theta}\ge20=ab$, which is (1). Hence P→2, Q→3, R→4, S→1. Numerical checks with random tangents confirmed the product $16$, the radius $5$, perpendicular slopes at a point on the circle $x^2+y^2=41$, and the minimum area $20$. The value (5) is a distractor.