Question

Let \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) and \( y = x + 1 \) intersect each other at points A & B, then \( \angle AOB \) (where O is the centre of the ellipse) is:

• A. \( \frac{\pi}{2} + \tan^{-1} \frac{1}{4} \)
• B. \( \frac{\pi}{2} + \tan^{-1} \frac{1}{3} \)
• C. \( \frac{\pi}{2} + \tan^{-1} \frac{1}{2} \)
• D. \( \frac{\pi}{4} + \tan^{-1} \frac{1}{2} \)

Hint:

When finding angles in ellipses and lines, use the properties of tangents and the geometry of the ellipse to simplify the calculations.

Answer 1

The Correct Option is A

Step 1: Equation of the Ellipse.
The equation of the ellipse is given as: \[ \frac{x^2}{2} + \frac{y^2}{1} = 1 \] which represents a standard ellipse with the center at the origin \( O(0, 0) \).
Step 2: Equation of the Line.
The equation of the line is given as \( y = x + 1 \), which is a straight line with slope 1.
Step 3: Find Points of Intersection (A and B).
To find the points of intersection, substitute \( y = x + 1 \) into the ellipse equation: \[ \frac{x^2}{2} + \frac{(x + 1)^2}{1} = 1 \] Expanding and solving this quadratic equation for \( x \), we get the values of \( x \) for the points of intersection \( A \) and \( B \).
Step 4: Use Geometry of Ellipse to Calculate \( \angle AOB \).
The angle \( \angle AOB \) can be calculated using the geometry of the ellipse. By using the properties of the ellipse and the equation of the line, the required angle is given by: \[ \angle AOB = \frac{\pi}{2} + \tan^{-1} \frac{1}{4} \] Final Answer: \[ \boxed{\frac{\pi}{2} + \tan^{-1} \frac{1}{4}} \]