Match each conic in List-I with its eccentricity in List-II. List-I (P) An ellipse whose latus rectum is half of its minor axis (Q) A hyperbola whose…

(P) For an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ the latus rectum is $\frac{2b^2}{a}$ and the minor axis is $2b$. Setting $\frac{2b^2}{a}=b$ gives $b=\frac a2$, so $e=\sqrt{1-\frac14}=\frac{\sqrt3}{2}$: P→4. (Q) Perpendicular asymptotes $y=\pm\frac bax$ require $b=a$, so $e=\sqrt{1+1}=\sqrt2$: Q→3. (R) The foci $(\pm ae,0)$ and the minor-axis ends $(0,\pm b)$ form a square exactly when $ae=b$; then $a^2e^2=a^2(1-e^2)$, so $e^2=\frac12$ and $e=\frac1{\sqrt2}$: R→1. (S) If a hyperbola and its conjugate have eccentricities $e_1,e_2$, then $\frac1{e_1^2}+\frac1{e_2^2}=1$. With $e_1=\frac53$: $\frac1{e_2^2}=1-\frac9{25}=\frac{16}{25}$, so $e_2=\frac54$: S→2. The remaining value $\frac23$ in item (5) is a distractor, obtained for example by mistakenly using the ellipse with $a=3,\ b^2=5$.