Question

Find the equation of the parabola whose vertex is at $(2,1)$ and the directrix is $x=y-1.$

Answer 1

The equation of parabola be $y+x+k=0$

It passes through $2,1$

$k=-(x+y)=-(2+1)=-3$

Solving $y+x-3=0$ and $x=y-1.$

We have $y=2,x=1$

$1=\frac{a+1}{2},a=3$

$2=\frac{b+1}{2},b=0$

Focus is $(3,0)$

Eq is $\sqrt{(x-3{)}^2+(y-0{)}^2}=\frac{x-y+1}{\sqrt{1^2+1^2}}$

Squaring and rearranging the terms we have,

$x^2+y^2-14x+2y+2xy+17=0$