The Line x-2y+12=0 is tangent to a circle at (0,6). The line y=2x-3 is tangent to the same circle at (3,3). Find the center of the circle.

Answer 1

The two equations each perpendicular to the tangents are

y = - 2x + 6 and y = - (1/2) x 9/2.

They intersect at (1,4)

Answer 2

Here are the directions.

Find the equation of the line perpendicular to the line (x-2y+12=0) which also passes through the point (0,6).

Similarly, find the equation of the line perpendicular to the line (y=2x-3) which also passes through the point (3,3). The point of intersection of those two lines will be the center of the circle. This is because lines/rays passing through the center of a circle (a radius for example) and a point of tangency are perpendicular to the tangent line at the point of tangency.

Both perpendiculars described above must pass through the center of the circle, so find the point of intersection to find that center.