Don't Solve for X

Geometry Level 2

How many times do the these two curves intersect?

Equation 1: x 2 + y 2 + 4 x + 6 y = 12 x^2 + y^2 + 4x + 6y = 12

Equation 2: 5 y = 4 x + 13 5y = 4x + 13

once twice three times they don't intersect

Matilda Belanno by Matilda Belanno , 40, USA

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2 solutions

Clearly, they intersect twice! Clearly, they intersect twice!

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Roger Erisman
Oct 9, 2015

First equation is circle. Second is straight line. They will intersect 0, 1(tangent), or 2 (secant) times. No other choices. Completing square on first equation yields C(-2, -3) and radius r = 5. Easy to determine that (-2,2), (3,-3), and (-7,-3) are points on circle. Putting each x value into line equation gives (-2,1) inside circle, (3,5) outside circle and (-7,-3) on circle. Therefore line intersects twice.

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