Write an equation

Question

The x- and y-coordinates of the focus of a parabola are (4,2) and the directrix is x=-8. A second parabola is a translation 3 units right and 1 unit up of the first parabola.Write an equation for the second parabola.

Answer 1

The second parabola focus is (4+3, 2+1) =(7, 3).

The directrix of the second parabola is moved 3 units to the left. The equation of the directrix is (x=-5).

 

Let (x, y) is a point on the second parabola. The distance between (x, y) and (7, 3) is the same as the distance between (x, y) and the directrix (x=-5).

 

The distance between (x, y) and (7, 3) is

        √[(x-7)²+(y-3)²] .................................(1)

 

The distance between (x, y) and (x=-5) is

x -(-5) ........................................................(2)

 

For a parabola the distances (1) and (2) are equal

 

√[(x-7)²+(y-3)²] = x -(-5)

 

We square both sides and obtain

(x-7)²+(y-3)²=(x+5)²

x²-14x+49+y²-6y+9=x²+10x+25

We move all variables and numbers on one side of the equation and obtain

-24x +y²-6y+33=0.

 

If we solve for x, the parabola is

x =1/24y²-1/4y+11/8.