for the conic y=5x^2-40x+78 find an equation in standard form and its vertex, focus, and directrix

Question

this is URGENT! I need it solved for tomorow!

Answer 1

Given equation y = 5x^2-40x+78

Add 2 to each side.

y+2 = 5x^2-40x+80

y+2 = 5(x^2-8x+16)

Divide to each side by 5.

(y+2)/5 = (x-4)^2

4/4(1/5)(y+2) = (x-4)^2

4/20(y+2) = (x-4)^2

4(1/20)(y-(-2)) = (x-4)^2

(x-4)^2 = 4(1/20)(y-(-2))

Compare it standard form of parabola (x-h)^2 = 4a(y-k)

a = 1/20, h = 4, k = -2.

Vertex = (h,k) = (4,-2)

Focus = (h,k+a) = (4,-2+1/20)

(4,-39/20)

Directrix = y-k+a = 0

y-(-2)+1/20 = 0

y+2+1/20 = 0

y = -2-1/20

y = -41/20