Question

The equation of 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

The correct option is D

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

Let P(x,y) be any point on the parabola whose focus is S(1,-1) and the directrix is x+y+7=0

Draw PM perpendicular to x+y+7=0

Then, we have : SP=PM

$\Rightarrow S{P}^2=P{M}^2$

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

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

$\Rightarrow 2(x^2+1-2x+y^2+1+2y)=x^2+y^2+49+2xy+14y+14x$

$\Rightarrow (2x^2+2-4x+2y^2+2+4y)=x^2+y^2+49+2xy+14y+14x$

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

Hence, the required equation is $x^2+y^2-2xy-18x-10y-45=0$