Question

The cosine of the acute angle between the curves $y=\left|x^2-1\right|$ and $y=\left|x^2-3\right|$ at their points of intersection is

• A. $\frac{1}{3}$
• B. $\frac{7}{9}$
• C. $\frac{11}{9\sqrt{2}}$
• D. $\frac{2}{7}$

Answer 1

Answer: (B)

For points of intersection $P\&Q$ , solving the curves $\left|x^2-1\right|=\left|x^2-3\right|,$ we get,
$\Rightarrow x^2-1=3-x^2$
$\Rightarrow x=\pm \sqrt{2}\Rightarrow P\left(\sqrt{2},1\right)\&Q\left(-\sqrt{2},1\right)$
For the slope of tangents at $x=\sqrt{2}$
$C_1$ is $y=x^2-1$
$\frac{dy}{dx}=2x$
$\Rightarrow m_1=2\sqrt{2}$
And $C_2$ is $y=3-x^2$
$\frac{dy}{dx}=-2x$
$\Rightarrow m_2=-2\sqrt{2}$
$\therefore tan\theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$=\left|\frac{4\sqrt{2}}{1-8}\right|=\frac{4\sqrt{2}}{7}$
$\therefore cos\theta =\frac{7}{9}$