If the line x-2y=12 is tangent to the ellipse x^2a^2+y^2b^2=1 at the point (3,-92), then t…

Question

If the line $x-2y=12$ is tangent to the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ at the point $\left(3,-\dfrac92\right)$, then the length of the latus rectum of the ellipse is:

Accepted Answer

Line: $\dfrac x{12}-\dfrac y6=1$. Tangent at $(3,-\frac92)$: $\dfrac{3x}{a^2}-\dfrac{4.5y}{b^2}=1$.

Matching: $\dfrac3{a^2}=\dfrac1{12}\Rightarrow a^2=36$; $\dfrac{4.5}{b^2}=\dfrac16\Rightarrow b^2=27$.

$$\text{LR}=\frac{2b^2}a=\frac{54}6=9.$$

Solutions · 1