Question

The angle of intersection of ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and circle \( x^2 + y^2 = ab \) is:

• A. \( \tan^{-1}\left( \frac{a + b}{ab} \right) \)
• B. \( \tan^{-1}\left( \frac{a - b}{ab} \right) \)
• C. \( \tan^{-1}\left( \frac{a + b}{\sqrt{ab}} \right) \)
• D. \( \tan^{-1}\left( \frac{a - b}{\sqrt{ab}} \right) \)

Hint:

When calculating the angle between two curves, find the derivatives at the point of intersection and use the formula for the tangent of the angle between two lines.

Answer 1

The Correct Option is B

Step 1: Equation of the ellipse and circle.
The equation of the ellipse is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] The equation of the circle is: \[ x^2 + y^2 = ab \]

Step 2: Calculate the slopes at the points of intersection.
At the points of intersection, we find the derivatives of both equations with respect to \( x \) and \( y \). For the ellipse: \[ \frac{d}{dx} \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right) = 0 \quad \implies \quad \frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0 \] Solving for \( \frac{dy}{dx} \) gives: \[ \frac{dy}{dx} = -\frac{b^2 x}{a^2 y} \] For the circle: \[ \frac{d}{dx} (x^2 + y^2) = 0 \quad \implies \quad 2x + 2y \cdot \frac{dy}{dx} = 0 \] Solving for \( \frac{dy}{dx} \) gives: \[ \frac{dy}{dx} = -\frac{x}{y} \]

Step 3: Angle of intersection.
The angle \( \theta \) of intersection between two curves is given by: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] where \( m_1 \) and \( m_2 \) are the slopes of the two curves at the point of intersection. Substitute the values of \( m_1 \) and \( m_2 \) from the previous step: \[ \tan \theta = \left| \frac{-\frac{b^2 x}{a^2 y} - \left( -\frac{x}{y} \right)}{1 + \left( -\frac{b^2 x}{a^2 y} \right) \cdot \left( -\frac{x}{y} \right)} \right| \] Simplifying this expression gives: \[ \tan \theta = \left| \frac{\frac{a^2 - b^2}{a^2}}{\frac{a^2 + b^2}{a^2}} \right| = \tan^{-1}\left( \frac{a - b}{\sqrt{ab}} \right) \]

Step 4: Conclusion.
Thus, the angle of intersection is: \[ \tan^{-1}\left( \frac{a - b}{\sqrt{ab}} \right) \] The correct answer is option (B).