The length of the latusrectum of the parabola 13x-32+y-42=2x-3y+52

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The length of the latusrectum of the parabola 13 $[{(x - 3)}^{2} + {(y - 4)}^{2}] = {(2 x - 3 y + 5)}^{2}$

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$\frac{1}{\sqrt{13}}$

$\frac{2}{\sqrt{13}}$

$\frac{5}{13}$

$\frac{4}{\sqrt{3}}$

answer is A.

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Detailed Solution

$G i v e n p a r a b o l a 13 [{(x - 3)}^{2} + {(y - 4)}^{2}] = {(2 x - 3 y + 5)}^{2} \Rightarrow {(x - 3)}^{2} + {(y - 4)}^{2} = (\frac{2 x - 3 y + 5}{\sqrt{13}}) I t i n t h e f o r m o f {SP}^{2} = {PM}^{2} N o w LLR = 4 a = 2 (p e r p e n d i c u l a r distance from focus (3, 4) to directrix 2 x - 3 y + 5 = 0) = 2 (\frac{|6 - 12 + 5|}{\sqrt{13}}) = \frac{2}{\sqrt{13}}$