Find the equation of the parabola with focus (2, 3) and directrix y = -1 ?

Question

Find the equation of the parabola with focus (2, 3) and directrix y = -1

Answer 1

y = ax^2

directrix ==> y = -1/4a

If directrix  y = -1

    

           -1 =  -1/4a

Multiply by -4a

4a = 1

a = 1/4

The standard form of the parabola is

y = ax^2  ==> y = 1/4x^2

Answer 2

Parabola is the locus of points that are equidistant to the focus and the directrix.

If M(x,y) is a point on the parabola, then

- The square of its distance to the focus F(2,3) is:

MF² = (x -- 2)² + (y -- 3)²

- The square of its distance MH to the directrix y = -1 is:

MH² = (y + 1)²

Since MF = MH

(x -- 2)² + (y - 3)² = (y + 1)²

(x -- 2)² =  (y + 1)² -- (y -- 3)²

x² -- 4x + 4 = (y + 1 + y -- 3)(y + 1 -- y + 3)

x² -- 4x + 4 = 8y -- 8  

8y = x² -- 4x + 12

ANSWER:

y = (1/8)x² -- (1/2)x + 3/2

Answer 3

Focal length = 2

vertex = (2,1)

(x - 2)² = 8(y - 1)

x² - 4x + 4 = 8y - 8

x² - 4x - 8y + 12 = 0