Question

The focus of a parabola is (4,7) and the directrix is y=8.



What is an equation of the parabola?

(x−4)2=−4(y−9)

(x−4)2=−2(y−7.5)

(x−4)2=−0.5(y−7.5)

(x−4)2=−(y−9)

Answer 1

step 1
we need to find the distance between  (x, y) and the focus (4,7)
(x, y)-------> is any point on the parabola.
d=√[(x-4)²+(y-7)²]

step 2
Find the distance between (x, y)  and the directrix y=8
the distance is given by
d=║(y-8)║

step 3
Equating the two expressions gives
√[(x-4)²+(y-7)²]=║(y-8)║
Squaring both sides give
(x-4)²+(y-7)²=(y-8)²
Expand and simplify
x²-8x+16+y²-14y+49=y²-16y+64
x²-8x+65=-2y+64
Complete the square  Remember to balance the equation 
x²-8x+16-16=-2y-1
(x²-8x+16)=-2y-1+16

Rewrite as perfect squares

(x-4)²=-2y+15
(x-4)²=-2(y-15/2)------->(x-4)²=-2(y-7.5)

the answer is
(x−4)2=−2(y−7.5)