Question

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

• A. Intersect at right angles at $(2,3)$
• B. Touch each other at $(2,3)$
• C. Do not intersect at $(2,3)$
• D. Intersect at an angle $\frac{\pi }{3}$

Answer 1

The correct option is B Touch each other at $(2,3)$

$y=x^2-1=8x-x^2-92x^2-8x+8=0x^2-4x+4=0{(x-2)}^2=0x=2y=x^2-1={\left(2\right)}^2-1=3$

So, curves intersect at $(2,3)$

Now, $m_1=\frac{dy}{dx}$ of $(x^2-1)$ at $(2,3)$

$m_1=2x=2\left(2\right)=4$

$m_2=\frac{dy}{dx}$ of $(8x-x^2-9)$ at $(2,3)$

$m_2=8-2x=8-2\left(2\right)=4$

Since $m_1=m_2$

we can say that curves touch each other at $(2,3)$.