If two curves touch each other at some point, find value of $\frac{2 \sqrt{3} \alpha}{5 \pi}$

Question

If two curves $y=2 \sin \frac{5 \pi}{6}x$ and $y=\alpha x^2-3 \alpha x+2\alpha +1$ touch each other at some point then find the value of $\frac{2 \sqrt{3} \alpha}{5 \pi}$? where $\bigg( 0 \leq x \leq \frac{18}{5} \bigg )$

Could someone give me hint as how to approach this question? I am not able to initiate.

Answer 1

In the polynomial, $y(1) = 1$ regardless of $\alpha$. In the trig function $y(1) = 1$, too. This means that the point of tangency, (if there is one) will be at (1,1).

Find $\frac {dy}{dx}$ for both functions. Then find $\alpha$ such that the slopes of the curves are equal when $x=1$

Answer 2

If Y is difference function

$$Y(x) = 2 \sin ( 5 \pi x/6) - \alpha x^2 + 3 \alpha x- 2 \alpha x -1 $$

in above, by inspection the point of intersection is $(x,y)=(1,1) $

$$ Y^{\prime}(x)= 5/3 * \cos( 5 \pi x/6) - 2 \alpha x + 3 \alpha = 0 $$

$$ Y^{\prime}(1)= (5/3) (-\sqrt3/2) + \alpha =0 $$

Find $\alpha$ and plug in... $ \rightarrow 1/ \pi$