Question

If a variable tangent to the curve x2y = c3 makes intercepts a, b on x and y axis respectively, then the value of a2b is

Answer 1

a²b = 27c³/4

find slope of tangent to the curve, x²y = c³ at point (h, k). where we assume (h,k) is a point on given curve.

so, 2xy + x²dy/dx = 0.

⇒dy/dx = -2xy/x² = -2y/x

at (h,k) slope of tangent = -2k/h

so, equation of tangent which is passing through (h,k) is ...

(y - k) = -2k/h(x - h)

⇒(y - k) + 2k/h(x - h) = 0

x - intercept = a ,

means, y = 0 and x = a⇒(0 - k) + 2k/h(a - h) = 0

⇒k = 2k/h(a - h)

⇒3h/2 = a ......(1)

similarly, y - intercept = b

means, x = 0, y = b ⇒(b - k) + 2k/h(0 - h) = 0

⇒b - k = 2k/h × h = 2k

⇒b = 3k ......(2)

now, a²b = (3h/2)² × (3k) = 27h²k/4

since, (h, k) is lies in the curve

so, h²k = c³

then, a²b = 27c³/4

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