Choose the correct answer. The equation of the ellipse whose focu… - Vidyadip

MCQ

Choose the correct answer. The equation of the ellipse whose focus is (1, -1), the directrix the line x - y - 3 = 0 and eccentricity $\frac{1}{2}$ is:

✓
$7 x^2+2 x y+7 y^2-10 x+10 y+7=0$
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

✓

Answer

Correct option: A.

$7 x^2+2 x y+7 y^2-10 x+10 y+7=0$

$7 x^2+2 x y+7 y^2-10 x+10 y+7=0$

Solution:
Given that, foums of the ellipse is $S(1,-1)$ and the equation of directrix is $x-y-3=0$
Also, $e=\frac{1}{2}$
From definition of ellipse, for any point $P(x, y)$ on the ellipse, we have $S P=e P M$, where $M$ is foot of the perpendicular from point $P$ to the directrix.
$\therefore \sqrt{(x-1)^2+(y+1)^2}=\frac{1}{2} \frac{|x-y-3|}{\sqrt{2}}$
$\Rightarrow 8 x^2-16 x+16+8 y^2+16 y=x^2+y^2+9-2 x y+6 y-6 x$
$\Rightarrow 7 x^2+2 x y+7 y^2-10 x+10 y+7=0$

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