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Find the angle between the curves xy = 6 and x2y =12.

Solution :

The equation of the two curves are

xy = 6          …….(i)

and, x2y = 12            …………(ii)

from (i) , we obtain y = 6x. Putting this value of y in (ii), we obtain

x2 (6x) = 12 6x = 12

x = 2

Putting x = 2 in (i)  or (ii), we get y = 3.

Thus, the two curves intersect at P(2, 3).

Differentiating (i) with respect to x, we get

xdydx + y = 0 dydx = yx

m1 = (dydx)(2,3) = 32

Differentiating (ii) with respect to x, we get

x2 dydx + 2xy  = 0 dydx = 2yx

m2 = (dydx)(2,3) = -3

Let θ be the angle, then angle between angle between two curves

tanθ = m_1 – m_2\over 1 + m_1 m_2 = 3\over 11

\theta = tan^{-1} (3/11)


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