Question

Show that if the point with coordinates (x,y) is equidistant from the point (2,0) and the line y=−4, then y=1/8(x-2)²-2.

Answer 1

The demonstration is showed below

Explanation:

The distance betwenn two points is given by:

`d = √((x2-x1)^2+(y2-y1)^2)`

If the point is equidistant from a point and a line, the distance must be equal. For the line let's select the point (x,-4), because the distance will be ortogonal, and is the small distance between a point and a line. So:

`√((x-2)^2+(y-0)^2)  = √((x-x)^2+(y+4)^2)`

Removing the squares:

(x-2)² + y² = (y+4)²

(x-2)² + y² = y² + 8y + 16

y² - y² - 8y = 16 - (x-2)²

8y = (x-2)² - 16

y = (1/8)*(x-2)² - 16/8

y = (1/8)*(x-2)² - 2