Question

The axis of a parabola is $y=x$, and the vertex and focus are at a distance $\sqrt{2}$ and $2\sqrt{2}$ respectively from the origin. Then, 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 $(x-y{)}^2=8(x+y-2)$

REF.Image.

Given parabola $y=x$

vertex distance $\sqrt{2}$ from origin

Let vertex $=(1,1)$

$a$ $b$

focus distance $2\sqrt{2}$ from origin

focus $=(2,2)$

$h$ $k$

Equation of directrix $\to x+y=0$

Equation of parabola $\sqrt{(x+h{)}^2+(y-k{)}^2}=\frac{ax+by+c}{\sqrt{a^2+b^2}}$

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

$2[x^2+y^2+4+4-4x-4y]=(x+y{)}^2$

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

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

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