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Family of Curves

Source: 1995 National High School Mathematics League, Exam Two, Problem 1

March 3, 2020
parameterization

Problem Statement

Give a family of curves 2(2sin⁡θ−cos⁡θ+3)x2−(8sin⁡θ+cos⁡θ+1)=02(2\sin\theta-\cos\theta+3)x^2-(8\sin\theta+\cos\theta+1)=0, where θ\theta is a parameter. Find the maximum value of the length of the chord that y=2xy=2x intersects the curve.
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