Finding Intersections of Circle and Line

Algebra 2 · Axiom Academy

EXAMPLE Finding Intersections of x^2+y^2=25 and y=2x-1 Use the substitution method to find exactly where a line crosses a circle. Find where the line y=2x-1 intersects the circle x^2+y^2=25 , using the substitution method. The circle x^2+y^2=25 , the line y=2x-1 , and the two points where they meet. Nice work — you found where a line and circle intersect using substitution. The pieces worth keeping: Substitution is key: solve the line equation for one variable, then substitute it into the circle equation. Expect a quadratic: substituting a line into a circle equation always yields a quadratic in one variable. The discriminant tells the story: positive means two intersections (secant), zero means one (tangent), negative means none (misses entirely). Find both coordinates: after solving for x , substitute back into the LINE equation to get each matching y . Round at the end, not mid-way: compute with the exact radical values and round only the final answer — rounding too early can shift the last digit. This method works for any line and circle — substitute, solve the resulting quadratic, and read the discriminant to know how many intersections to expect.

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