Question

the line with equation y=kx intersects the circle with equation x^2-10x+y^2-12y+57=0 at two distinct points

Answer 1

x^2 - 10x + y^2 - 12y + 57 = 0

x^2 - 10x + 25 + y^2 - 12x + 36 = -57 + 25 + 36

(x - 5)^2 + (y - 6)^2 = 4

(x - 5)^2 + (y - 6)^2 = 2^2

This is the equation of the circle.
The line y = kx is a line that passes through the point (0, 0).
Between two values of k, t intersects the circle at 2 points.
The values of k are the calues that make the lines y = kx tangents to the circle.