Question

The two curves $y=3^x$ and $y=5^x$ intersect at an angle

• A. ${\tan }^{-1}\left(\frac{\log 3-\log 5}{1+\log 3\log 5}\right)$
• B. ${\tan }^{-1}\left(\frac{\log 3+\log 5}{1-\log 3\log 5}\right)$
• C. ${\tan }^{-1}\left(\frac{\log 3+\log 5}{1+\log 3\log 5}\right)$
• D. ${\tan }^{-1}\left(\frac{\log 3-\log 5}{1-\log 3\log 5}\right)$

Answer 1

Answer: (A)

Given curves $y=3^x$ ...(i)
and $y-5^x$ ...(ii)
Intersect at the point $(0,1)$.
Now, differentiating Eqs. (i) and (ii) w.r.t. $x$, we get
$\frac{dy}{dx}=3^x\log 3$ and $\frac{dy}{dx}=5^x\log 5$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{(0,1)}=\log 3$ and ${\left(\frac{dy}{dx}\right)}_{(0,1)}=\log 5$
$\Rightarrow m_1=\log 3$ and $m_2=\log 5$
Angle between these curves is given by
$\tan \theta =\frac{m_1-m_2}{1+m_1{m}_2}$
$\Rightarrow \tan \theta =\frac{\log 3-\log 5}{1+\log 3\cdot \log 5}$
$\Rightarrow \theta ={\tan }^{-1}\left(\frac{\log 3-\log 5}{1+\log 3\log 5}\right)$