Question

What is the equation of the parabola with a focus at (7,5) and a directrix of y= -3?

Answer 1

Parabola's equation is `y=1/16(x-7)^2+1` and vertex is `(7,1)`.

Explanation:

Parabola is locus of a point which moves so that its distance from a given point calld focus and a given line ccalled directrix is always constant.

Let the point be `(x,y)`. Here focus is `(7,5)` and distance from focus is `sqrt((x-7)^2+(y-5)^2)`. Its distance from directrix `y=-3` i.e. `y+3=0` is `|y+3|`.

Hence equaion of parabola is

`(x-7)^2+(y-5)^2)=|y+3|^2`

or `x^2-14x+49+y^2-10y+25=y^2+6y+9`

or `x^2-14x+65=16y`

i.e. `y=1/16(x^2-14x+49-49)+65/16`

or `y=1/16(x-7)^2+(65-49)/16`

or `y=1/16(x-7)^2+1`

Hence parabola's equation is `y=1/16(x-7)^2+1` and vertex is `(7,1)`.

graph{(1/16(x-7)^2+1-y)((x-7)^2+(y-1)^2-0.15)((x-7)^2+(y-5)^2-0.15)(y+3)=0 [-12.08, 27.92, -7.36, 12.64]}