2788.31 – Two Circles and a Line


Tags:Problem Set 99Geometry-Analytical

Here are two concentric circles with center at the origin. The equation of the bigger circle is x2+y2=25x^2 + y^2 = 25. The straight line AC cuts the bigger circle at A and C and the smaller circle at B (2,1).

What's the equation of line AC?


Solution

We go with y=mx+by = mx + b, passing through the points (2,1) and (-5,0).

0=5m+b0 = -5m + b and 1=2m+b1=7mm=17 1 = 2m + b \rightarrow 1 = 7m\rightarrow m = \frac{1}{7}.

0=57+bb=570 = \frac{-5}{7} + b \rightarrow b = \frac{5}{7}

So y=17x+57y = \frac{1}{7}x + \frac{5}{7}