Length of the latus rectum of the parabola $25 [(x - 2)^{2} + (y - 3)^{2}] = (3 x - 4 y + 7)^{2}$ is :

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1/5

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2/5

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Solution

The correct option is D 2/5
$T h e g i v e n e q u a t i o n i s 25 [{(x - 2)}^{2} + {(y - 3)}^{2}] = {(3 x - 4 y + 7)}^{2} H e r e, f o c u s o f p a r a b o l a (2, 3) D i r e c t r i x 3 x - 4 y + 7$

$N o w, w e k n o w d i s t a n c e b e t w e e n d i r e c t r i x a n d f o c u s i s h a l f o f l a t u s r e c t u m . s o, d i s t a n c e b e t w e e n t h e m \Rightarrow L = ∣ ∣ ∣ ∣ \frac{3 \times 2 - 4 \times 3 + 7}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣ L = ∣ ∣ ∣ \frac{6 - 12 + 7}{\sqrt{25}} ∣ ∣ ∣ ∴ L = ∣ ∣ ∣ \frac{1}{5} ∣ ∣ ∣ T h e n l a t u s r e c t u m = 2 L = 2 \times \frac{1}{5} = \frac{2}{5}$

$H e n c e, t h e o p t i o n D i s t h e c o r r e c t a n s w e r .$

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