Question

The equation to the parabola whose focus is $(1,-1)$ and the directrix is $x+y+7=0$ is

• A. $x^2+y^2-2xy-18x-10y=0$
• B. $x^2-18x-10y-45=0$
• C. $x^2+y^2-18x-10y-45=0$
• D. $x^2+y^2-2xy-18x-10y-45=0$

Answer 1

Answer: (D)

$x^2+y^2-2xy-18x-10y-45=0$

Given that:

Focus$\left(f\right)=(1,-1)$ and eqn of the directrix $:x+y+7=0$

We know that eccentricity of the parabola is $1$ i.e. distance from a fixed point and a fixed line are equal.

$\therefore \sqrt{(x-1{)}^2+(y+1{)}^2}=\left|\frac{x+y+7}{\sqrt{1+1}}\right|$

Squaring on both sides, we get

$(x-1{)}^2+(y+1{)}^2=\frac{|x+y+7{|}^2}{2}$

$⟹2x^2-4x+2+2y^2+4y+2=x^2+y^2+49+2xy+14y+14x$

$⟹x^2+y^2-2xy-18x-10y-45=0$

Therefore, Equn. of required parabola is $x^2+y^2-2xy-18x-10y-45=0$

Hence, D is the correct option.