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Q.
If the vertex of the parabola is the point $(-3, 0)$ and the directrix is the line $x + 5 = 0$, then its equation is
Conic Sections
Solution:
Given, vertex $\left(-3,0\right)$ and directrix $x + 5 = 0$
Let the coordinate of focus be $S\left(a, 0\right)$.
We know, vertex is the mid-point of point of intersection of directrix with axis and focus.
$\therefore \, \left(-3, 0\right)=\left(\frac{-5+a}{2}, 0\right)$
$\Rightarrow \, -3=\frac{-5+a}{2}$
$\Rightarrow a=-1$
$\therefore $ Coordinate of focus is $\left( - 1,0\right)$.
By definition of parabola, $PM^{2} = PS^{2}$
$\therefore \, \left(\frac{x+5}{\sqrt{1}}\right)^{2}=\left(x+1\right)^{2}+\left(y-0\right)^{2}$
$\Rightarrow \, x^{2} + 25 +10x- x^{2} -1 - 2 x - y ^{2} = 0$
$\Rightarrow \, y^{2} = 8\left(x + 3\right)$ is the required equation.
