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VITEEE 2010 Paper

Question : 117 of 120

Marks: +1, -0

y=3^{x}

y=5^{x}

intersect at an angle

Solution:

Given curves are

y=3^{x}

...(i)

and

y=5^{x}

...(ii)

intersect at the point (0,1).

Now, differentiating eqs. (i) and (ii) w.r.t.x, we get

\frac{dy}{dx}=3^{x}\log3

and

\frac{dy}{dx}=5^{x}\log5

⇒

\left[\frac{dy}{dx}\right]_{(0,1)}=\log3

and

\left[\frac{dy}{dx}\right]_{(0,1)}=\log5

⇒

m_{1}=\log3

and

m_{2}=\log5

Angle between these curves is given by

\tan\theta=\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}

⇒

\tan\theta=\frac{\log3-\log5}{1+\log3\cdot\log5}

⇒

\theta=\tan^{-1}\left(\frac{\log3-\log5}{1+\log3\log5}\right)

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