Question

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

• A. $\left|\frac{lna-lnb}{1+lnalnb}\right|$
• B. $\left|\frac{lna-lnb}{1-lnalnb}\right|$
• C. $\left|\frac{lna+lnb}{1+lnalnb}\right|$
• D. $\left|\frac{lna+lnb}{1-lnalnb}\right|$

Answer 1

The correct option is A $\left|\frac{lna-lnb}{1+lnalnb}\right|$

Point of intersection of both the curve is $(0,1)$
Now let $c_1:y=a^x,and{c}_2:y=b^x$
${\left(\frac{dy}{dx}\right)}_{c_1}=m_1=a^x\log a$
${\left(\frac{dy}{dx}\right)}_{c_2}=m_2=b^x\log b$
Thus at (0,1) $m_1=\log a,m_2=\log b$
If angle of intersection of both the curves is
$\alpha$ then, $\tan \alpha =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{\log a-\log b}{1+\log a\log b}\right|$