Circle? conic sections?

Question

find an equation of the line passing through the center of the circle

16x^2 + 64x +16y - 32y = 119

and the vertex of the parabola (x-3)^2 = 4(y-5)

i need solutions. thanks!

Answer 1

It seems you have made a mistake in writing the equation of the circle. I assume it is

16x^2 + 64x +16y^2 - 32y = 119.

Its center is (-32, 16)

Vertex of the parabola (x-3)^2 = 4(y-5) is

(3, 5)

Slope of the line through (-32, 16) and (3, 5) is

m = (16 - 5) / (-32 -3) = - 11/35

Equation of the line having slope m = - 11/35 and passing through (3, 5) is

y - 5 = (-11/35) (x - 3)

=> 11x + 35y = 208.