The line L is a tangent to the circle x2 + y2 = 45 at the point (–3, 6). The line L crosses the x-axis at the point P. Work out the coordinates of P.

Question

i’m very confused with this question 

Answer 1

Line L is perpendicular to the radius that goes through the point of tangency and the center of the circle, (0,0).

 

      Slope of line through (-3,6) and (0,0):    (0-6) / (0-(-3)) = -2

 

      So, slope of L is -1/(-2) = 1/2.

 

      L has slope 1/2 and contains the point (-3,6).  So, equation of L is y-6 = (1/2)(x-(-3))

 

                                                                                                      y-6 = (1/2)x + 3/2

 

                                                                                                         y = (1/2)x + 15/2

 

       Since point P is on the x-axis, its y-coordinate is 0.  So, 0 = (1/2)x + 15/2

 

                                                                                        x = -15

 

       P = (-15,0)