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Q.

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

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a

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

b

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

c

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

d

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

answer is A.

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$\begin{array}{l} P M^{2} = P S^{2} \Rightarrow (x - 5)^{2} + (y - 3)^{2} = {(\frac{3 x - 4 y + 1}{\sqrt{9 + 16}})}^{2} \\ \Rightarrow 25 (x^{2} + 25 - 10 x + y^{2} + 9 - 6 y) \\ = 9 x^{2} + 16 y^{2} + 1 - 24 x y - 8 y + 6 y \\ \Rightarrow 16 x^{2} + 9 y^{2} - 256 x - 142 y + 24 x y + 849 = 0 \\ \Rightarrow (4 x + 3 y)^{2} - 256 x - 142 y + 849 = 0 \end{array}$

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