Question

What is the equation of the parabola with a focus at (13,16) and a directrix of y= 17?

Answer 1

`(x-13)^2=-2(y-33/2)`

Explanation:

Use

Distance of ( x, y ) from the focus ( 13, 16 )

= Distance from the directrix y = 17.

`sqrt((x-13)^2+(y-16)^2) = 17-y`, giving

`(x-13)^2=-2(y-33/2)`

Note that the size of the parabola, a = 1/2

See the second graph, for clarity, by suitable scaling.

The vertex is in the proximity of directrix and the focus is just below,

graph{((x-13)^2+2(y-33/2))(y-17)((x-13)^2+(y-16)^2-.01)=0 [0, 25, 0, 20]}

graph{((x-13)^2+2(y-33/2))(y-17)((x-13)^2+(y-16)^2-.001)=0 [10, 16, 14, 18]}