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Find the equation of the family of orthogonal trajectories of the parabolas y = a*x² (a being a parameter).

1. x²/2 + y² = c
2. x² + y²/2 = c
3. x²/2 - y² = c
4. x² - y²/2 = c

Correct answer: x²/2 + y² = c

Solution

Eliminating a gives dy/dx = 2y/x; replacing the slope with its negative reciprocal yields x dx + 2y dy = 0, whose integral is x²/2 + y² = c, a family of ellipses.

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