Question

The equation of the parabola with focus (0, 0) and directrix x + y = 4 is 

Options

• x² + y² − 2xy + 8x + 8y − 16 = 0 

•  x² + y² − 2xy + 8x + 8y = 0

• x² + y² + 8x + 8y − 16 = 0 

•  x² − y² + 8x + 8y − 16 = 0 

Answer 1

x² + y² − 2xy + 8x + 8y − 16 = 0 

Let P (x, y) be any point on the parabola whose focus is S (0, 0) and the directrix is x + y= 4. 

 

Draw PM perpendicular to x + y = 4.
Then, we have: \[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 0 \right)^2 + \left( y - 0 \right)^2 = \left( \frac{x + y - 4}{\sqrt{2}} \right)^2 \]
\[ \Rightarrow x^2 + y^2 = \left( \frac{x + y - 4}{\sqrt{2}} \right)^2 \]
\[ \Rightarrow 2 x^2 + 2 y^2 = x^2 + y^2 + 16 + 2xy - 8x - 8y\]
\[ \Rightarrow x^2 + y^2 - 2xy + 8x + 8y - 16 = 0\]