The equation of parabola whose focus is $(5, 3)$ and directrix is $3 x - 4 y + 1 = 0$ , is

(4 x + 3 y)^{2} - 256 x - 142 y + 849 = 0

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(4 x - 3 y)^{2} - 256 x - 142 y + 849 = 0

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(3 x + 4 y)^{2} - 142 x - 256 y + 849 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(3 x - 4 y)^{2} - 256 x - 142 y + 849 = 0

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Solution

The correct option is A $(4 x + 3 y)^{2} - 256 x - 142 y + 849 = 0$
Let $P (h, k)$ be the point on required parabola
By definition
$S P = P M$
$\sqrt{{(5 - h)}^{2} + {(k - 3)}^{2}} = \frac{| 3 h - 4 k + 1 |}{\sqrt{{(3)}^{2} + {(4)}^{2}}}$
$\Rightarrow$ $25 {(5 - h)}^{2} + 25 {(k - 3)}^{2} = {(3 h - 4 k + 1)}^{2}$
$\Rightarrow$ $16 h^{2} - 256 h + 9 k^{2} - 142 k + 24 h k + 849 = 0$
$\Rightarrow$ ${(4 h + 3 k)}^{2} - 256 h - 142 k + 849 = 0$

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