Question

Obtain the equation of the parabola with focus (3,2) and directrix 3x-4y+9=0?

Answer 1

`16x^2+9y^2+24xy-204x-28y+244=0`

Explanation:

You suppose a point on the parabola `(x,y)`

The distance between the point `(x,y)` and the directrix is the same distance from the point `(x,y)` to the focus

`sqrt((x-3)^2+(y-2)^2)=|\3x-4y+9|/sqrt(3^2+(-4)^2)`

now by squaring both sides

`(x-3)^2+(y-2)^2=(3x-4y+9)^2/25`

Simplify

`25(x^2-6x+9+y^2-4y+4)=(3x+9)^2+(2)(-4y)(3x+9)+16y^2`

now multiply and simplify you get the following

`16x^2+9y^2+24xy-204x-28y+244=0`