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

For the line y=kx to intersect the circle
`x^2-10x+y^2-12y+57=0`at two distinct points, the discriminant of the quadratic equation obtained by substituting y=kx into the circle's equation must be positive. Solving this inequality yields the range of possible values for k: 0.71<k<2.15.

Consider the line y=kx and the circle
`x^2-10x+y^2-12y+57=0`. For the line to intersect the circle at two distinct points, the quadratic equation obtained by substituting y=kx into the circle's equation should have two real and distinct roots.

This quadratic equation is
`(1+k^2)x^2-(10+12k)x+57=0.` The discriminant of this quadratic is responsible for determining the nature of its roots.

For two distinct roots, the discriminant must be greater than zero. This leads to the inequality
`21k^2-60k+32 < 0`, which can be solved to obtain the range of possible values for k: 0.71<k<2.15.

Complete question below:

The line with equation y=kx intersects the circle with equation
`x^2-10x+y^2-12y+57=0` at two distinct points. Determine the range of possible values for k.