Question

If the two curves $y=a^x$ and $y=b^x$ intersect at an angle $\alpha$ , then tan a equals

• A. $\frac{\log a-\log b}{1+\log a\log b}$
• B. $\frac{\log a+\log b}{1-\log a\log b}$
• C. $\frac{\log a-\log b}{1-\log a\log b}$
• D. None of these

Answer 1

Answer: (A)

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 $P(0,1)$.
$\therefore m_1={\left(\frac{dy}{dx}\right)}_p={\left(a^x\log a\right)}_{(0,1)}=\log a$
$m_2={\left(\frac{dy}{dx}\right)}_p={\left(b^x\log b\right)}_{(0,1)}=\log b$
If $\alpha$ be the angle of intersection of the tangents.
Then, $\tan \alpha =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$\Rightarrow \tan \alpha =\mid \frac{\log a-\log b}{1+\log -a\log b}\mid$