In a rectangular coordinate system let F0 4 be a given point

In a rectangular coordinate system, let F(0, 4) be a given point; let y = -8 be an equation defining the line L; and let P(x, y) be any other point. Further, let denote the distance between the two points f and P, and d_2 denote the distance between the point P and the line L. Find an equation which defines the curve that consists of all P P(x, y) such that d_1 = d_2?. Find the x intercepts of the curve, if they exist. Find the y-intercepts of the curve, if they exist.

Solution

F (0, 4) ; line L: y = -8 ;

Point P (x , y)

d1 = distance between F and P : sqrt{( x^2 +(y-4)^2}

d2 = distance between F and P = (y +8)/sqrt(1) = y+8

d1 = d2

sqrt{( x^2 +(y-4)^2} = y+8

squaring both sides:

(x^2 +( y- 4)^2 = (y +8)^2

x^2 +y^2 +16 -8y = y^2 +64 +16y

x^2 = 24y +48

a) x^2 = 24(y +2)

b) x intercepts : plug y=0 : x^2 = 48

x = + /- 4sqrt3

c) y intercept : plug x=0

y = -2