Intersecting GraphsDate: 10/22/96 at 20:27:21 From: Geoff Peters Subject: Intersecting Graphs Hi Dr. Math! I'm an age 15 grade 10 student living in B.C. Canada, and I'm wondering if you could help me with this "bonus" math problem. (In this question the ^ denotes an exponent). Intersecting graphs The graph of x^2 + xy + y^2 = 3 intersects the graph of x + xy + y + 1 = 0 in k distinct points. What is the value of k? Thanks so much for your time. I'd really appreciate it if you could give me an answer to that question! (My whole math class has not been able to get the correct answer.) Date: 10/24/96 at 16:49:44 From: Doctor Rob Subject: Re: Intersecting Graphs The graphs of the two equations intersect at simultaneous solutions of the two equations. First we will solve the second equation: 0 = x + xy + y + 1 = (x + 1)(y + 1) This tells us that all solutions of this equation will have either x = -1 or else y = -1 (or both). Now we solve each of these together with the first equation: x = -1 and x^2 + xy + y^2 = 3 imply that 1 - y + y^2 = 3 or that y^2 - y - 2 = 0 = (y - 2)(y + 1) so y = -1 or y = 2. This gives two points, (x, y) = (-1, -1) and (-1, 2). y = -1 and x^2 + xy + y^2 = 3 imply that 1 - x + x^2 = 3, or that x^2 - x - 2 = 0 = (x - 2)(x + 1), so x = -1 or x = 2. This gives two points, (x, y) = (-1, -1) and (2, -1). There are three solutions, namely (-1, -1), (-1, 2), and (2, -1). Thus k = 3. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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