Find the equation of the parabola whose focus | vedclass

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(A) Let $P\ (x, y)$ be any point on the parabola with focus $S\ (-1, -2)$ and directrix $x - 2y + 3 = 0$.By the definition of a parabola,the distance

Vedclass · Accepted Answer

$4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0$

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(A) Let $P\ (x, y)$ be any point on the parabola with focus $S\ (-1, -2)$ and directrix $x - 2y + 3 = 0$.By the definition of a parabola,the distance from $P$ to the focus $S$ is equal to the perpendicular distance from $P$ to the directrix.$SP = PM$$SP^2 = PM^2$$(x + 1)^2 + (y + 2)^2 = \left( \frac{|x - 2y + 3|}{\sqrt{1^2 + (-2)^2}} \right)^2$$(x^2 + 2x + 1) + (y^2 + 4y + 4) = \frac{(x - 2y + 3)^2}{5}$$5(x^2 + y^2 + 2x + 4y + 5) = x^2 + 4y^2 + 9 - 4xy + 6x - 12y$$5x^2 + 5y^2 + 10x + 20y + 25 = x^2 + 4y^2 - 4xy + 6x - 12y + 9$$4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0$

Find the equation of the parabola whose focus is $(-1, -2)$ and whose directrix is the line $x - 2y + 3 = 0$.

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