If the line 3x-2y+12=0 intersects the parabola 4y=3x^2 at the points A and B, then at the …

Question

If the line $3x-2y+12=0$ intersects the parabola $4y=3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to:

Accepted Answer

$y=\dfrac{3x+12}{2}$. Substituting into $4y=3x^2$: $2(3x+12)=3x^2 \Rightarrow x^2-2x-8=0\Rightarrow x=-2,4$.

$A=(-2,3),\ B=(4,12)$. Slopes from origin: $m_{OA}=-\dfrac32,\ m_{OB}=3$.

$$\tan\theta=\left|\frac{-3/2-3}{1-9/2}\right|=\frac{9}{7} \;\Rightarrow\; \theta=\tan^{-1}\left(\frac97\right).$$

Solutions · 1