Question

The equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 is

Options

•  x² + y² − 2xy − 18x − 10y = 0 

•  x² − 18x − 10y − 45 = 0 

•  x² + y² − 18x − 10y − 45 = 0 

•  x² + y² − 2xy − 18x − 10y − 45 = 0

   

Answer 1

x² + y² − 2xy − 18x − 10y − 45 = 0 

Let P (x, y) be any point on the parabola whose focus is S (1, −1) and the directrix is x + y+ 7 = 0. 

 

Draw PM perpendicular to x + y + 7 = 0.
Then, we have: 

\[SP = PM\]
\[ \Rightarrow S P^2 = P M^2 \]
\[ \Rightarrow \left( x - 1 \right)^2 + \left( y + 1 \right)^2 = \left( \frac{x + y + 7}{\sqrt{1 + 1}} \right)^2 \]
\[ \Rightarrow \left( x - 1 \right)^2 + \left( y + 1 \right)^2 = \left( \frac{x + y + 7}{\sqrt{2}} \right)^2 \]
\[ \Rightarrow 2\left( x^2 + 1 - 2x + y^2 + 1 + 2y \right) = x^2 + y^2 + 49 + 2xy + 14y + 14x\]
\[ \Rightarrow \left( 2 x^2 + 2 - 4x + 2 y^2 + 2 + 4y \right) = x^2 + y^2 + 49 + 2xy + 14y + 14x\]
\[ \Rightarrow x^2 + y^2 - 45 - 10y - 2xy - 18x = 0\] 

Hence, the required equation is x² + y² − 2xy − 18x − 10y − 45 = 0.