Question

If $(0,4)$ and $(0,2)$ are respectively the vertex and focus of a parabola, then its equation is

• A. $x^2+8y=32$
• B. $y^2+8x=32$
• C. $x^2-8y=32$
• D. $y^2-8x=32$

Answer 1

The correct option is A $x^2+8y=32$

Since $(0,4)$ is the vertex and $(0,2)$ is the focus of the parabola, the parabola is a downward parabola.

Its equation is ${(x-0)}^2=4a(y-4)$

i.e. $X^2=4aY[X=x-0,Y=y-4]$

Its focus is given by $X=0,Y=a$

i.e. $x=0,y-4=a$ i.e. $y=4+a$

But the given focus is $(0,2).$

$\therefore 4+a=2\Rightarrow a=-2$

$\therefore$ Required parabola is ${(x-0)}^2=4(-2)(y-4)$ i.e. $x^2=-8(y-4)=-8y+32$

$\Rightarrow x^2+8y=32$