Question

Find the equation of the parabola whose: 

 focus is (1, 1) and the directrix is x + y + 1 = 0

Answer 1

Let P (x, y) be any point on the parabola whose focus is S (1, 1) and the directrix is x+ y + 1 = 0.
Draw PM perpendicular to x + y + 1 = 0.
Then, we have: 

$$SP=PM$$
$$\Rightarrow S{P}^2=P{M}^2$$
$$\Rightarrow {(x-1)}^2+{(y-1)}^2={\left|\frac{x+y+1}{\sqrt{1+1}}\right|}^2$$
$$\Rightarrow {(x-1)}^2+{(y-1)}^2={\left(\frac{x+y+1}{\sqrt{2}}\right)}^2$$
$$\Rightarrow 2(x^2+1-2x+y^2+1-2y)=x^2+y^2+1+2xy+2y+2x$$
$$\Rightarrow (2x^2+2-4x+2y^2+2-4y)=x^2+y^2+1+2xy+2y+2x$$
$$\Rightarrow x^2+y^2-2xy-6x-6y+3=0$$