How to find the equation of a parabola given with the given focus and directrix?

Question

F(1,0)

x=-4

Answer 1

For the given parabola directrix is x = - 4 and focus at (1, 0)

The required parabola is horizontal. Since directrix equation is in the form of x = h - p

Standard form of the horizontal parabola is ( y – k )² = 4p ( x – h )

Where Center (h, k ) , focus is at (h +p, k ) and directrix is x  = h - p

 

Directrix equation x = h – p

Actual directrix is x = - 4

h - p =  - 4              -------------------> (1)

 

Focus is (h + p, k)

Actual focus is (1 , 0)

h + p = 1                -------------------> (2)

and k = 0

 

Add the equations (1) & (2).

(1) + (2)      ⇒    ( h – p ) + ( h + p )  =  - 4 + 1      ⇒     2h = -3

h = -3/2

Substitute h = -3/2  in equation (2).

(-3/2) + p = 1

p = 1 + (3/2)

p = 5/2

Substitute h, k , p values in standard form of parabola.

(y-0)² =4(5/2)(x-(-3/2))

y² = 10 ( x +3/2 )

Solution : Parabola equation is y² = 10 ( x +3/2 ).