Question

The point P = (4, 3) lies on the circle x2 + y2 = 25. Find an equation for the line that is tangent to the circle at P. This line meets the x-axis at a point Q. Find an equation for the other line through Q that is tangent to the circle, and identify its point of tangency

Answer 1

If the center is at O, then the slope of OP = 3/4

So, the tangent line, which is perpendicular to the radius OP has slope -4/3, and its equation is thus

y-3 = -4/3 (x-4)

The line's x-intercept is at Q=(7,0)

Due to symmetry, the other tangent line through Q touches the circle at (-3,4), so its equation is

y+3 = 4/3 (x-4)

see

http://www.wolframalpha.com/input/?i=plot+x%5E2%2By%5E2%3D25,+y+%3D+-4%2F3+(x-4)%2B3,+y+%3D+4%2F3+(x-4)-3,+y%3D0