If the curves , intersect each other at right angles, then the value of is
- A.6
- B.
- C.4
- D.
Show answer and solution
Correct option: D
Solution
At a point of intersection, the slopes of the two curves must satisfy m₁·m₂ = -1.
From y² = 6x: differentiate to get 2y(dy/dx) = 6, so dy/dx = 3/y.
From 9x² + by² = 16: differentiate to get 18x + 2by(dy/dx) = 0, so dy/dx = -9x/(by).
For orthogonality: (3/y)·(-9x/(by)) = -1 → 27x/(by²) = 1 → by² = 27x.
But on the curve y² = 6x, so b(6x) = 27x → 6b = 27 → b = 27/6 = 9/2.
From y² = 6x: differentiate to get 2y(dy/dx) = 6, so dy/dx = 3/y.
From 9x² + by² = 16: differentiate to get 18x + 2by(dy/dx) = 0, so dy/dx = -9x/(by).
For orthogonality: (3/y)·(-9x/(by)) = -1 → 27x/(by²) = 1 → by² = 27x.
But on the curve y² = 6x, so b(6x) = 27x → 6b = 27 → b = 27/6 = 9/2.
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