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

Answer 1

The correct option is A $y^2=8(x+3)$


We have vertex $(-3,0)$ and foot of directrix $(-5,0)$
Clearly axis of parabola is $x-$axis as both vertex and foot of directrix lie on $x-$axis
$\Rightarrow a=2$
[Distance between vertex and foot of directrix]
Since directrix lies on left side of vertex, hence focus will lie on right side of vertex.
Hence focus is at same distance as between vertex and foot of directrix and lies on $x$ axis
Hence, focus is $(-1,0)$
Let $P(x,y)$ is any variable point on parabola.
$\Rightarrow \sqrt{(x+1{)}^2+y^2}=|(x+5)|$
Squaring on both sides
$\Rightarrow (x+5{)}^2=(x+1{)}^2+y^2$
$\Rightarrow x^2+10x+25=x^2+1+2x+y^2$
$\Rightarrow y^2=8x+24$
$\Rightarrow y^2=8(x+3)$

Hence, option $\left(A\right)$ is correct.