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Match the items of Column – I with Column – II
Column – I Column – II
I The length of the latus rectum of the hyperbola $16 x^{2} - 9 y^{2} = 144$ is P $\frac{2}{3}$
II The product of the perpendiculars drawn from any point on the hyperbola $x^{2} - 2 y^{2} = 2$ to its asymptotes is Q 3
III The length of the transverse axis of the hyperbola $x y = 18$ is R $\frac{32}{3}$
IV The product of the lengths of the perpendiculars drawn from the foci of $3 x^{2} - 4 y^{2} = 12$ on any of its tangents is S 12
T 5

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$I - R; I I - P; I I I - S; I V - Q$

$I - P; I I - Q; I I I - R; I V - S$

$I - P; I I - Q; I I I - S; I V - T$

$I - Q; I I - R; I I I - T; I V - P$

answer is A.

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Detailed Solution

I. If y the parabola a is $\frac{x^{2}}{2} - \frac{y^{2}}{1} = 1$
length of the latus rectum $= \frac{2 b^{2}}{a} = \frac{32}{3}$
II. the hyperbola is is $\frac{x^{2}}{2} - \frac{y^{2}}{1} = 1$
Required product $= \frac{a^{2} b^{2}}{a^{2} + b^{2}} = \frac{2}{3}$
III. Given hyperbola is xy =18
length of the latus - rectum $= 2 \sqrt{2 C} = 12$
IV. the hyperbola is $\frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$
Required product $= b^{2} = 3$ .

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