Mk y the subject of the equation
100x-500y-5x^2-5y^2=0
100x-500y-5x^2-5y^2=0
or
5y^2 + 500y - 100x+5x^2 = 0
y^2 + 100y - 20x+x^2
consider this to be a quadratic equation with variable y, so that
a = 1
b= 100
c = (x^2 - 20x)
then y = (-100 ± √(100^2 - 4(1)(x^2 - 20x) ) ) /2
= (-100 ± √10000-4x^2 + 80x)/2
= (-100 ± 2√(2500 - x^2 + 20x) )/2
= -50 ± √(2500 + 20x - x^2)
or
recognize it is a circle:
5x^2 - 100x + 5y^2 + 500y = 0
x^2 - 20x + ... + y^2 + 100y + ... = 0
x^2 - 20x + 100 + y^2 + 100y + 2500 = 0 + 100 + 2500
(x-10)^2 + (y+ 50)^2 = 2600
(y+50)^2 = 2600 - (x+10)^2 = 2600 - x^2 + 20x - 100
(y+50)^2 = 2500 + 20x - x^2
y + 50 = ± √(2500+20x-x^2
y = -50 ± √(2500+20x-x^2)