Question

The curves $y=x^2-1$ and $y=8x-x^2-9$

• A. Intersect at right-angle
• B. Touch each other
• C. Do not intersect
• D. Intersect at an angle $\frac{\pi }{3}$

Answer 1

The correct option is B Touch each other

By solving the two curve equations we will get the point of intersection of the curves.
$\Rightarrow x^2-1=8x-x^2-9$
$\Rightarrow 2x^2-8x+8=0$
$\Rightarrow x^2-4x+4=0$
$\Rightarrow (x-2{)}^2=0$
$\Rightarrow x=2,2$
$\Rightarrow y=x^2-1=3$
hence both the curves intersect each other at $(2,3)$
$\Rightarrow$ Slope of tangent at $(2,3)$ on $y=x^2-1$
$\Rightarrow \frac{dy}{dx}=2x$
$\Rightarrow m_1={\left(\frac{dy}{dx}\right)}_{(2,3)}=4$
Now, Slope of tangent at $(2,3)$ on $y=8x-x^2-9$
$\Rightarrow \frac{dy}{dx}=8-2x$
$\Rightarrow m_2={\left(\frac{dy}{dx}\right)}_{(2,3)}=4$
$\{\because m_1=m_2=4\text{at}(2,3\left)\right\}$
$\Rightarrow$ Curves touch each other at $(2,3)$