Question

The axis of a parabola is along the line $y=x$ and the distance of the origin from its vertex is $\sqrt{2}$ and that from its focus is $2\sqrt{2}$ respectively. If the vertex and focus both lie in the first quadrant then the equation of the parabola is

• A. ${(x+y)}^2=(x-y-2)$
• B. ${(x-y)}^2=(x+y-2)$
• C. ${(x-y)}^2=4(x+y-2)$
• D. ${(x-y)}^2=8(x+y-2)$

Answer 1

The correct option is D ${(x-y)}^2=8(x+y-2)$

Distance of vertex from origin$=\sqrt{2}$
Distance of focus from origin$=2\sqrt{2}$

Axis is along $y=x$.

The focus and the vertex lie in the first quadrant. Hence, the coordinates of the focus are $(2,2)$.

The vertex is at $(1,1)$

Hence, the equation of directrix is $x+y=0$

Take point $(h,k)$ on the parabola. By definition
${(h-2)}^2+{(k-2)}^2={\left|\frac{h+k}{\sqrt{2}}\right|}^2$

On squaring,
$h^2-8h+8+k^2-8k+8=2hk$

Substitute $(h,k)$ to $(x,y)$

${(x-y)}^2=8(x+y-2)$

Hence, option D is correct.