Q.245. Length of common tangent to two curves.

Question 245.
There is one line which is tangent to the curve y = 1/x at some point A and at the same time tangent to the curve y = x^2 at some point B. What is the distance between A and B?

Answer 245.
Let A(a, 1/a) and B(b, b^2) be the contact points of the common tangent to the two curves.

y = x^2 => dy/dx = 2x
=> dy/dx at B = 2b
=> the eqn. of tangent to the curve y = x^2 at B is
y - b^2 = 2b(x - b)
=> y = 2bx - b^2

Solving this with the first curve y = 1/x should give two identical roots for it to be the tangent to it.
=> 2bx - b^2 = 1/x
=> 2bx^2 - b^2x - 1 = 0
For roots to be identical, discriminant = 0
=> b^4 + 8b = 0
=> b (b^3 + 8) = 0
=> b = 0 or - 2
=> B = (0, 0) or (-2, 4)

Also, the root of the quadratic eqn. is b^2/4b = b/4 = 0 or - 1/2
=> A = (- 1/2, - 2) ... [the root 0 cannot give any A]

=> AB^2 = (-2 + 1/2)^2 + (4 + 2)^2 = 9/4 + 36 = 153/4
=> AB = (1/2)√(153).

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