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Polar Coordinates Problem

Source: 1984 National High School Mathematics League, Exam One, Problem 3

February 22, 2020
analytic geometryconicshyperbola

Problem Statement

For any integers 1≤n≤m≤51\leq n\leq m\leq5, how many hyperbolas does the equation ρ=11−Cmncos⁡θ\rho=\frac{1}{1-\text{C}_m^n \cos\theta} represent? Note: Cmn=m!n!(m−n)!\text{C}_m^n=\frac{m!}{n!(m-n)!}. (A)15(B)10(C)7(D)6\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6
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