Question

Axis of a parabola is $y=x$ and vertex and focus are at a distance $\sqrt{2}$ and $2\sqrt{2}$ respectively from the origin. The equation of the parabola is

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

Answer 1

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

Given that

Axis of parabola is $x=y$

$\Rightarrow$Vertex is at distance $\sqrt{2}$ from origin and lies on the axis

Therefore, $\Rightarrow vertex=(1,1)$

$\Rightarrow$ Focus is at a distance $2\sqrt{2}$ from origin and lies on the axis.

Therefore, $\Rightarrow vertex=(2,2)$

So, we can say that equation of $Directrix\to x+y=0$

Hence equation of parabola is

$\Rightarrow (x-2{)}^2+(y-2{)}^2=(\frac{x+y}{\sqrt{2}}{)}^2$

$\Rightarrow x^2-4x+4+y^2-4y+4=\frac{x^2+y^2+2xy}{2}$

$\Rightarrow 2x^2-8x+8+2y^2-8y+8=x^2+y^2+2xy$

$\Rightarrow x^2+y^2-2xy=8x+8y-16$

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