# Find the Equation of the Parabola Whose: Focus is (1, 1) and the Directrix is X + Y + 1 = 0 - Mathematics

Find the equation of the parabola whose:

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

#### Solution

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

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

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.1 | Q 1.2 | Page 24