Question

Find the equation to the parabola with focus (a, b) and directrix $\frac{x}{a}+\frac{y}{b}=1.$

Answer 1

Let $P(x,y)$ be a point on the parabola with the directrix M:$\frac{x}{a}+\frac{y}{b}=1$ and focus S:(a,b)

we know that ,$S{P}^2=P{M}^2$

$(x-a{)}^2+(y-b{)}^2$ =${\frac{(\frac{x}{a}+\frac{y}{b}-1)}{\frac{1}{a^2}+\frac{1}{b^2}}}^2$

After rearranging the terms in both sides,we get

$a^2{x}^2+b^2{y}^2+a^2{y}^2+b^2{x}^2-2a^{3}x-2yb{a}^2-2ax{b}^2-2y{b}^3+a^4+2a^2{b}^2+b^4$ =

$a^2{y}^2+b^2{x}^2+a^2{b}^2+2abxy-2b^{2}xa-2a^{2}yb$

After simplifying both sides,we get

$(ax-by{)}^2-2a^{3}x-2b^{3}y+a^4+a^2{b}^2+b^4=0$