Answer to Question #234701 in Math for Johnnie

Question

Establish the equation of the parabola with focus (-3,-4) and directrix 5x+6y-5=0

Answer 1

A parabola is defined as follows:

For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.

Let $P(x,y)$ be the point on the parabola. Then

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

$$61x^2+366x+549+61y^2+488y+976$$

$$=25x^2+36y^2+25+60xy-50x-60y$$

$$36x^2-60xy+25y^2+416x+548y+1500=0$$

Therefore the required equation of the parabola is

$$(6x-5y)^2+416x+548y+1500=0$$