Question

Find the equation of the parabola whose: 

 focus is (0, 0) and the directrix 2x − y − 1 = 0

 

Answer 1

Let P (x, y) be any point on the parabola whose focus is S (0, 0) and the directrix is 2x− y − 1 = 0.
Draw PM perpendicular to 2x − y − 1 = 0.
Then, we have: 

\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 0 \right)^2 + \left( y - 0 \right)^2 = \left| \frac{2x - y - 1}{\sqrt{4 + 1}} \right|^2 \]
\[ \Rightarrow x^2 + y^2 = \left( \frac{2x - y - 1}{\sqrt{5}} \right)^2 \]
\[ \Rightarrow 5 x^2 + 5 y^2 = 4 x^2 + y^2 + 1 - 4xy + 2y - 4x\]
\[ \Rightarrow x^2 + 4 y^2 + 4xy - 2y + 4x - 1 = 0\]