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Q.
If the two curves $ y= a^{x} $ and $ y = b^{x} $ intersect at an angle $ \alpha $ , then tan a equals
ManipalManipal 2008
Solution:
The angle between the two curves is the angle between their tangents at their point of contact.
$\therefore $ The point of intersection of two curves is $a^{x}=b^{x}$
For distinct values of a and $b$, the equality is true, if $x=0$
Now, if $x=0$, then $y=a^{0}=1$
$\therefore $ The two curves intersect at $(0,1)$.
Let $m_{1}$ and $m_{2}$ be the slope of the tangent to the curve $y=a^{x}$ and $y=b^{x}$ respectively at $P(0,1)$.
$\therefore m_{1}=\left(\frac{d y}{d x}\right)_{p}=\left(a^{x} \log a\right)_{(0,1)}=\log a$
$m_{2}=\left(\frac{d y}{d x}\right)_{p}=\left(b^{x} \log b\right)_{(0,1)}=\log b$
If $\alpha$ be the angle of intersection of the tangents.
Then, $\tan \alpha=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|$
$\Rightarrow \tan \alpha=\mid \frac{\log a-\log b}{1+\log -a \log b}\mid$