Question

If the vertex of the parabola is the point $(-3,0)$ and the directrix is the line $x+5=0$, then its equation is

• A. $y^2=8(x+3)$
• B. $x^2=8(y+3)$
• C. $y^2=-8(x+3)$
• D. $y^2=8(x+5)$

Answer 1

Answer: (A)

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, $P{M}^2=P{S}^2$
$\therefore {\left(\frac{x+5}{\sqrt{1}}\right)}^2={\left(x+1\right)}^2+{\left(y-0\right)}^2$
$\Rightarrow x^2+25+10x-x^2-1-2x-y^2=0$
$\Rightarrow y^2=8\left(x+3\right)$ is the required equation.