Ellipse
Ellipse
nta_abhyas_2025
Grade 11
Question:
The line $2x + y = 3$ cuts the ellipse $4x^2 + y^2 = 5$ at points $P$ and $Q$. If $\theta$ is the acute angle between the normals at $P$ and $Q$, then $\theta$ is equal to
\cos^{-1}\left(\frac{7}{25}\right)
\sin^{-1}\left(\frac{24}{25}\right)
\cos^{-1}\left(\frac{3}{5}\right)
\cot^{-1}\left(\frac{7}{3}\right)
Step-by-Step Solution
Key Concept: The area of a triangle formed by tangent lines to an ellipse and the coordinate axes can be computed using the coordinates of intersection points and the formula for tangent slopes.
From the equation of the ellipse $y = 3 - 2x$ and $4x^2 + y^2 = 5$, we find intersection points $P(1,2)$ and $Q(1,1)$. The slopes of tangents at $P$ and $Q$ are $-1c$ and $-4$ respectively. Using the tangent slope formula for an ellipse and the angle between two lines, we calculate $\cot\theta = 1$ which gives $\theta = \frac{\pi}{4}$. The area of triangle $OAB$ formed by the tangents and axes is $\frac{1}{2} \times 6 \times 8 = 24$ square units.
Correct Answer: 24