Question

Find the equation of the parabola whose focus is $(-1,1)$ and directrix is $x+y+1=0$

Answer 1

Parabola is the locus of all points whose distance from the focus is equal to the distance from directrix.

Let $(x,y)$ be a point on the parabola, then

Distance from focus is: $\sqrt{(x+1{)}^2+(y-1{)}^2}$

Distance from directrix is: $\frac{|x+y+1|}{\sqrt{2}}$

So, we have:

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

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

$x^2+y^2+2x-6y-2xy+3=0$

Thus the above equation represents the required parabola.