The equation of the ellipse whose equation of directrix is $3 x + 4 y - 5 = 0$ , coordinates of the focus are $(1, 2)$ and the eccentricity is $\frac{1}{2}$ is

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Solution

Let $P (x, y)$ be any point on the ellipse and PM be the perpendicular from P upon the directrix $3 x + 4 y - 5 = 0$ .
Then by the definition,
$\frac{S P}{P M} = e$

$S P = e . P M$
$\sqrt{(x - 1)^{2} + (y - 2)^{2}} = \frac{1}{2} | \frac{3 x + 4 y - 5}{\sqrt{3^{2} + 4^{2}}} |$

$(x - 1)^{2} + (y - 2)^{2} = \frac{1}{4} . \frac{(3 x + 4 y - 5)^{2}}{25}$

$100 (x^{2} + y^{2} - 2 x - 4 y + 5) = 9 x^{2} + 16 y^{2} + 24 x y - 30 x - 40 y + 25$
$91 x^{2} + 84 y^{2} - 24 x y - 170 x - 360 y + 475 = 0$ is the equation of the ellipse.

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