Systems of Conics

Algebra 2 · Axiom Academy

Where two curves cross is where their equations agree — solving a system of conics means finding the points the graphs share. 1. The Solutions Are Where the Graphs Meet Graph both equations on one plane. A point sitting on both curves makes both equations true at once — so the solution set of the system is precisely the set of intersection points. Here the circle x^2+y^2=25 and the line y=x+1 cross at two places, giving two solutions. The system — a circle and a line Its solutions — the two intersection points 2. Collapse the System to One Equation For exact answers, get rid of a variable. Substitute the line y=x+1 into the circle and the two-variable system becomes a single quadratic in x . Its roots are exactly the x -coordinates of the intersection points; back-substitute into the line to recover each y . Substitute y=x+1 into the circle Expand — the system collapses to one quadratic Factor to get the two x -coordinates Back-substitute into the line for each y 3. How Many Times Can They Cross? Slide a line toward a circle. Far away it misses — no real solution. At the instant it just touches, there is exactly one: a tangent , where the collapsed quadratic has a double root. Push through and there are two, a secant . The number of solutions is simply the number of real roots of the collapsed equation. 0 solutions — the line misses the circle; 25-k^2 is negative, so there is no real root. 1 solution — the line is tangent; 25-k^2=0 , a single double root.

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