Question

The equation of parabola whose focus is at $(4,-3)$ and the vertex is $(4,-1)$ is given by

• A. $x^2-8x+8y+24=0$
• B. $y^2-8y+8x+24=0$
• C. $x^2-4x+8y+24=0$
• D. $x^2-x+4y+20=0$

Answer 1

The correct option is A $x^2-8x+8y+24=0$

The vertex and focus of parabola lie on its axis.

Points $(4,-3)$ and $(4,-1)$ lie on the line $x=4$

So, the axis of parabola is the line $x=4$

Focus of the parabola lies below the vertex. So, the parabola is downwards opening

So, the equation of parabola is $(x-4{)}^2=-4a(y+1)$

Distance between focus and vertex $=-1-(-3)=2$

$\therefore a=2$

Vertex of parabola is $(4,-1)$

So, the equation of parabola is $(x-4{)}^2=-4(2\left)\right(y+1)$

$⟹x^2-8x+16+8y+8=0$

$⟹x^2-8x+8y+24=0$

So, the answer is option (A).