Math Help?!!?

Question

There is a graph of an ellipse centered at (0,0). f = (4,0). P = (x,y). N = ((25/4), y).

Let P(x,y) be a point such that PF/PN = (4/5). Use distance formula to write this equation in terms of x and y

Answer 1

Distance formula = √(Δx^2 + Δy^2)
Length of PF = √((x-4)^2 + (y-0)^2) = √((x-4)^2 + y^2)
Length of PN = √((x - 25/4)^2 + (y - y)^2) = √(x - 25/4)^2

Create a set of equal ratios:
PF/PN = 4/5
√(((x-4)^2 + y^2) / (x - 25/4)^2) = 4/5

Square both sides:
((x-4)^2 + y^2) / (x - 25/4)^2) = 16/25

Cross multiply:
25 ((x-4)^2 + y^2) = 16(x - 25/4)^2

Expand and simplify:
25 (x^2 - 8x + 16 + y^2) = 16(x^2 - 25/2x + 625/16)
25x^2 - 200x + 400 + 25y^2 = 16x^2 - 200x + 625

Final equation:
9x^2 + 25y^2 = 225