Could someone please help me with this math question?

Question

What is the equation of a parabola with (−2, 4) as its focus and y = 6 as its directrix?

 

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Thank you to anyone who helps!

Answer 1

the standard form of an equation of a parabola is:

 

(x-h)²=4p(y-k), where the focus is (h,k+p) and the directrix is y =k-p

 

since y = 6 is the directrix, the parabola opens down and its line of symmetry is parallel to the y axis

 

The focus is ( -2,4), ( h,k+p) , h=-2, and k+p=4 ...Eq 1

 

the directrix is y = 6 = k-p...Eq 2 , solving Eq 2, k = p+6

 

substituting in Eq 1 leads, p+6+p=4, 2p+6=4, 2p=-2, p = -1

Therefore k-1=4, k=5

 

The equation is:

 

(x-h)²=4p(y-k), ( x+2)² = 4(-1)(y-5), (x+2)² = -4 (y-5)

 

Expanding x²+4x+4=-4y+20, rearranging,

 

x²+4x+4y-16=0 is the general form of the equation

 

 

Answer 2

It helps if you sketched this out first. The vertex is halfway between the focus and the directrix.....which makes the vertex (-2,5). The distance from the vertex to focus, p=-1. This will be a parabola opening downward so

 

(x-h)²= 4p(y-k) is the general formula. so your answer will be

 

(x+2)² = -4(y-5)

 

Hope this helped.