Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8

Equation of the parabola is

Step-by-step explanation:

In this question the given focus is (2,4) and a directrix of y = 8 and we have to derive the equation of the parabola.

Let (x,y) is a point on the given parabola.Then the distance between the point (x,y) to (2,4) and the distance from (x,y) to diractrix will be same.

Distance between (x,y) and (2,4)

= √(x-2)²+(y-4)²

And the distance between (x,y) and directrix y=8

= (y-8)

Now √(x-2)²+(y-4)² = (y-8)

(x-2)²+(y-4)² = (y-8)²

x²+4-4x+y²+16-8y = y²+64-16y

x²+20+y²-4x-8y = y²-16y+64

x²+20-4x-8y+16y-64=0

x²+8y-4x-44 = 0

8y = -x²+4x+44

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.