Angle between two parabolas

Text

At what angle do the parabolas $y=2x^{2}+1$ and $y=x^{2}-x+3$ intersect?

Solution

Intersections: $2x^{2}+1=x^{2}-x+3\Rightarrow x^{2}+x-2=0\Rightarrow x=1,\ x=-2$, i.e. $A=(1,3)$ and $B=(-2,9)$. Slopes: $y_1'=4x$, $y_2'=2x-1$. The angle between two lines of slopes $m_1,m_2$ satisfies $\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|$.

• At $A$ ($x=1$): $m_1=4$, $m_2=1$, $\tan\theta=0.6\Rightarrow \theta\approx30.96^\circ$.
• At $B$ ($x=-2$): $m_1=-8$, $m_2=-5$, $\tan\theta=\left|\dfrac{-3}{41}\right|\approx0.0732\Rightarrow \theta\approx4.18^\circ$. $\boxed{\theta_A\approx30.96^\circ,\qquad \theta_B\approx4.18^\circ}$