The equation of the parabola having focus at (–1, –2) and the directrix x – 2y + 3 = 0 is __________.

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Solution

Let P(x, y) be any point in parabola
The distance between focus and P is $\sqrt{{(x - 1)}^{2} + {(y + 2)}^{2}} . . . (1)$
And distance (perpendicular distance) of directrix from point P is $|\frac{x - 2 y + 3}{\sqrt{1 + 4}}| . . . (2)$
By equating (1) and (2),
We get
$\sqrt{{(x + 1)}^{2} + {(y + 2)}^{2}} = |\frac{x - 2 y + 3}{\sqrt{5}}|$
By squaring both sides, we get
${(x + 1)}^{2} + {(y + 2)}^{2} = {|\frac{x - 2 y + 3}{\sqrt{5}}|}^{2} i . e . x^{2} + 2 x + 1 + y^{2} + 4 y + 4 = \frac{1}{5} [{(x - 2 y)}^{2} + 9 + 2 \times 3 (x - 2 y)] i . e . x^{2} + y^{2} + 2 x + 4 y + 5 = \frac{1}{5} [x^{2} + 4 y^{2} - 4 x y + 9 + 6 x - 12 y] i . e . 5 x^{2} + 5 y^{2} + 10 x + 20 y + 25 = x^{2} + 4 y^{2} - 14 x y + 9 + 6 x - 12 y i . e . 4 x^{2} + y^{2} + 14 x y + 4 x + 32 y + 16 = 0$
is the required equation of parabola

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