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Which of the following pairs of curves intersect orthogonally (c and k are arbitrary constants)?
- 16x² + y² = c and y¹⁶ = kx
- y = x + c*e^-x and x + 2 = y + k*e^-y
- x = c*x² and x² + 2y² = k
- x² - y² = c and xy = k
Correct answer: x² - y² = c and xy = k
Solution
For x² - y² = c, dy/dx = x/y; for xy = k, dy/dx = -y/x. Their product is -1, so these orthogonal-hyperbola families cut at right angles.
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