First slide

Ellipse

Question

$The equation of the ellipse having focus (1, - 1), directrix x - y - 3 = 0 and eccentricity 1 / 2 is$

Moderate

A

$7 x^{2} + 2 x y + 7 y^{2} - 10 x + 10 y + 7 = 0$
B

$7 x^{2} + 2 x y + 7 y^{2} + 7 = 0$
C

$7 x^{2} + 2 x y + 7 y^{2} + 10 x - 10 y - 7 = 0$
D

$None of these$

Solution

$Let P (x, y) be any point on the ellipse.$
$Then by definition, S P = e P M (where S is the focus and P M is$
$the distance of P from the directrix). So$
$\begin{array}{l} \sqrt{(x - 1)^{2} + (y + 1)^{2}} = \frac{1}{2} |\frac{x - y - 3}{\sqrt{2}}| \\ \Rightarrow (x - 1)^{2} + (y + 1)^{2} = \frac{1}{8} (x - y - 3)^{2} \end{array}$
$\Rightarrow 7 x^{2} + 2 x y + 7 y^{2} - 10 x + 10 y + 7 = 0$

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