2788.31 – Two Circles and a Line

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 ACAC cuts the bigger circle at AA and CC and the smaller circle at B=B = (2,1)$.

What's the equation of line ACAC?

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Solution

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

0=−5m+b0 = -5m + b and 1=2m+b→1=7m→m=171 = 2m + b \rightarrow 1 = 7m\rightarrow m = \dfrac{1}{7}.

0=−57+b→b=570 = \dfrac{-5}{7} + b \rightarrow b = \dfrac{5}{7}

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