Ellipse
Ellipse
nta_abhyas_2025
Grade 11
Question:
The point, which is at the shortest distance from the line $x + y = 7$ and lying on an ellipse $x^2 + 2y^2 = 6$, has coordinates $(a, b)$ then the value of $\frac{a}{b}$ is
Step-by-Step Solution
Key Concept: For tangent to ellipse parallel to a given line, use parametric form and equate slopes
The ellipse equation is $\frac{x^2}{6} + \frac{y^2}{2} = 1$ with $a^2 = 6$, $b^2 = 2$. Any point on the ellipse is $(\sqrt{6}\cos\theta, \sqrt{2}\sin\theta)$. The tangent at this point has equation $\frac{x\cos\theta}{\sqrt{6}} + \frac{y\sin\theta}{\sqrt{2}} = 1$. The tangent parallel to $x + y = 7$ (slope = -1) must have $\frac{\cos\theta}{\sqrt{6}} = \frac{\sin\theta}{\sqrt{2}}$, giving $\tan\theta = \frac{1}{\sqrt{3}}$. The point of tangency and tangent equation follow from this condition.
Correct Answer: 2