Question

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

(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 1

 Since, $(P S)^2=(P M)^2$

$\therefore  (x-5)^2+(y-3)^2=\left|\frac{3 x-4 y+1}{\sqrt{9+16}}\right|^2 \\$

$\Rightarrow  25\left(x^2+25-10 x+y^2+9-6 y\right) \\$

$=9 x^2+16 y^2+1^2-24 x y-8 y+6 x \\$

$\Rightarrow  16 x^2+9 y^2-256 x-142 y+24 x y+849=0 \\$

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

So, The correct option of this question will be (A).