Question

If a tangent having slope \(\frac{1}{3}\) to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b)\) is normal to the circle \((x+1)^2 + (y+1)^2 = 1\), then \(a^2\) lies in the interval?

• A. \(\left(\sqrt{\frac{2}{5}}, 2\right)\)
• B. \(\left(\frac{2}{5}, 4\right)\)
• C. \(\left(1, \frac{10}{9}\right)\)
• D. \((3,5)\)

Hint:

Combine ellipse tangent and circle normal conditions to find range for \(a^2\).

Answer 1

The Correct Option is B

The normal condition and tangent slope relate \(a^2, b^2\). Using normal to the circle and tangent slope to ellipse, solve inequalities to find \[ a^2 \in \left(\frac{2}{5}, 4\right). \]