Question

The equation of the parabola whose focus is (1, −1) and the directrix is x + y + 7 = 0 is
(a) x² + y² − 2xy − 18x − 10y = 0
(b) x² − 18x − 10y − 45 = 0
(c) x² + y² − 18x − 10y − 45 = 0
(d) x² + y² − 2xy − 18x − 10y − 45 = 0

Answer 1

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

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