Question

Find the equation to the parabola with focus (3, -4) and directrix 6x - 7y + 5 = 0.

Answer 1

Given the focus of the parabola as S:$(3,-4)$ and directrix M:$6x-7y+5$=0.

Let P$(x,y)$ be the point on the parabola,such that $S{P}^2$=$P{M}^2$.

$(x-3{)}^2+(y+4{)}^2$ = $\frac{(6x-7y+5{)}^2}{6^2+7^2}$

on simplifying both sides,we get

$(7x+6y{)}^2-216x-294x-60x+288y+492y+70y+(9\times 36+49\times 9+49\times 16+36\times 16)-25$=$0$

After adding alike terms,we get

$(7x+6y{)}^2-570x+750y+2100$=$0$

Therefore the required equation of the parabola is $(7x+6y{)}^2-570x+750y+2100$=$0$.