Ellipse & Parabola
Area of triangle from tangents to ellipse
MJAT_TS8_P2
Grade 12
Question:
Let $S(x_1,0)$ with $x_1>0$ be the focus of ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. Suppose a parabola whose vertex lies on the ellipse and whose axis is the $x$-axis touches the ellipse at $A$ (first quadrant) and $B$ (fourth quadrant). If tangents to the ellipse at $A$ and $B$ meet at $C$, then the area of $\triangle ABC$ is:
Step-by-Step Solution
Key Concept: Focus $S=(\sqrt{5},0)$. Vertex $V=(x_1,0)$ on ellipse. Parabola $y^2=-4a(x-x_1)$ touches ellipse. Substituting into ellipse equation: $x^2/9+(−4a(x-x_1)/4)=1$, i.e., $x^2/9-a(x-x_1)=1$. For tangency: discriminant $=0\Rightarrow$ finds $a$ and $x_1$.
Area $\approx\mathbf{15.08}$.
Correct Answer: 15.08