Question

The curves $y=x^2$ and $y=x^3$ intersect at an angle $\theta$ in the first quadrant. If $\tan \theta =m$, then the value of $\left|\frac{1}{m}\right|$ is

• A. 5
• B. 7
• C. $\frac{1}{5}$
• D. $\frac{1}{7}$

Answer 1

The correct option is B 7

Let the curves $y=x^2$ and $y=x^3$ intersect at $(\alpha ,\beta$

Then, $\beta ={\alpha }^2$ and $\beta ={\alpha }^3$

Thus, ${\alpha }^2={\alpha }^3$

Or, ${\alpha }^2(\alpha -1)=0$

Or, $\alpha =0,or\alpha =1$

Since the point is in the first quadrant, $\alpha =1$
$\beta =1$

Let the slope of the tangent of the curve $y=x^2$ be $m_1$ and that of $y=x^3$ be $m_2$

$m_1=\frac{d}{dx}\left(x^2\right){|}_{(1,1)}$

$=2xatx=1$

$=2$

$m_2=\frac{d}{dx}\left(x^3\right){|}_{(1,1)}$

$=3x^{2}atx=1$

$=3$

Thus, if $\theta$ is the angle of intersection of these curves, then
$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

Here, $\tan \theta =m$

$m=\left|\frac{3-2}{1+3\times 2}\right|$

$\Rightarrow m=\frac{1}{7}$

$\Rightarrow \left|\frac{1}{m}\right|=7$