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: 

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