Ellipse
Locus of pole of tangent — hyperbola eccentricity
MJAT_TS3_P2
Grade 12
Question:
If the variable line $y=kx+2h$ is a tangent to the ellipse $2x^2+3y^2=6$, then the locus of $P(h,\frac{5k}{2})$ is a conic $C$ whose eccentricity equals:
A) $\dfrac{3}{7}$
B) $\sqrt{\dfrac{3}{7}}$
C) $\sqrt{\dfrac{7}{3}}$
D) $2$
Step-by-Step Solution
Key Concept: Tangent condition to $\frac{x^2}{3}+\frac{y^2}{2}=1$: $c^2=3k^2+2$. With $c=2h$: $4h^2=3k^2+2\Rightarrow 4h^2-3k^2=2$.
Locus is a hyperbola. From $4h^2-3k^2=2$ and substitution: $e=\sqrt{\mathbf{7/3}}$ (option C).
Correct Answer: C