Find the length of latus rectum of the parabola whose focus is the point $(2, 3)$ and directrix is the line $x - 4 y + 3 = 0.$

\frac{14}{\sqrt{17}}

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\frac{17}{\sqrt{17}}

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\frac{15}{\sqrt{17}}

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\frac{10}{\sqrt{17}}

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Solution

The correct option is A $\frac{14}{\sqrt{17}}$
By definition the distance of any point on the parabola from the focus is equal to its distance from the directrix.
$∴ \sqrt{{{(x - 2)}^{2} + {(x - 3)}^{2}}} = \frac{x - 4 y + 3}{{(1 + 16)}^{1 / 2}},$
$17 (x^{2} + y^{2} - 4 x - 6 y + 13) = x^{2} + 16 y^{2} + 9 - 8 x y - 24 y + 6 x$
or $16 x^{2} + y^{2} + 8 x y - 74 x - 78 y + 212 = 0$
We know L.R. $= 4 a = 2 (2 a)$ where $2 a$ is the distance between the focus and directrix
$= 2 ∣ ∣ ∣ ∣ \frac{2.1 - 4.3 + 3}{{(1 + 16)}^{1 / 2}} ∣ ∣ ∣ ∣ = \frac{14}{\sqrt{17}} .$
Ans: A

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