a circle C has equation x^2 + y^2 - 6x + 10y + 9 =0 . The line with equation y=kx where k is a constant, cuts C at two distinct points. Find the range of values for k

Answer 1

x^2 -6x + 9 + y^2 +10y + 25 =25

(x-3)^2 + (y+5)^2 = 5^2

It's a circle with radius 5 and center at (3,-5). It's upper maximum point is (3,0), bottom minimum point is (3,-10). Point with max x value is (8,-5) and min x value is the point (-2,-5)

y=kx is a line that cuts the circle in 2 points

y=0 would just touch the circle at it's max point. k=0 is too small k>0

0<k<15/8 is where the line doesn't intersect the circle

what you want is k<0 or k>15/8

the point (-24/17,-45/17) and the point (3,0) are the two places on the circle where

a line from the origin just touches the circle at one point only.