Ellipse
Ellipse
nta_abhyas_2025
Grade 11
Question:
If the line $x - 2y = 12$ is a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $\left(3, -\frac{9}{2}\right)$, then the length of the latus rectum of the ellipse is
8 units
12y, 2 units
8y, 3 units
Step-by-Step Solution
Key Concept: Length of latus rectum of ellipse is $\frac{2b^2}{a}$; use point on ellipse to find parameters
The point $(3, -\frac{3}{2})$ lies on the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$. The equation of tangent at this point is $\frac{3x}{9} - \frac{3y/2}{4} = 1$. Simplifying: $\frac{x}{3} - \frac{3y}{8} = 1$. Comparing with given tangent $x - 2y - 12 = 0$ yields $a = 3$ and $b = \sqrt{3}$. Therefore, length of LR = $\frac{2b^2}{a} = \frac{2 \times 3}{3} = 9$ units.
Correct Answer: 3