Find the equation of ellipse when: Focus is $(1, 2)$ , directrix is $3 x + 4 y - 5 = 0$ and $e = \frac{1}{2}$ .

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Solution

Given: $e = \frac{1}{2}, S (1, 2)$ and equation of directrix is $3 x + 4 y - 5 = 0$

Let a point $P (x, y)$ , such that

$S P = e \cdot P M$ , where $P M$ is perpendicular distance from $P (x, y)$ to directrix

$\Rightarrow \sqrt{(x - 1)^{2} + (y - 2)^{2}} = \frac{1}{2} \times ∣ ∣ ∣ ∣ \frac{3 x + 4 y - 5}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣$

$\Rightarrow \sqrt{(x - 1)^{2} + (y - 2)^{2}} = \frac{1}{10} | 3 x + 4 y - 5 |$

Squaring on both sides, we get

$\Rightarrow (x - 1)^{2} + (y - 2)^{2} = \frac{1}{100} (3 x + 4 y - 5)^{2}$

$\Rightarrow 100 (x^{2} - 2 x + 1 + y^{2} - 4 y + 4)$

$= 9 x^{2} + 16 y^{2} + 25 + 24 x y - 40 y - 30 x$

$\Rightarrow 91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$

Hence, the equation of ellipse is

$91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$

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Similar questions

Find the equation of the ellipse in the following cases:
(i) focus is (0,1), directrix is x+y=0 and $e = \frac{1}{2}$ .
(ii) focus is (-1,1), directrix is x-y+3=0 and $e = \frac{1}{2}$ .
(iii) focus is (-2,3), directrix is 2x+3y+4=0 and $e = \frac{4}{5}$ .
(iv) focus is (1,2), directrix is 3x+4y-5=0 and $e = \frac{1}{2}$ .