Question

Consider ellipse \( E_k : \frac{x^2}{k} + \frac{y^2}{k} = 1 \), for \( k = 1, 2, \dots, 20 \). Let \( C_k \) be the circle which touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse \( E_k \). If \( r_k \) is the radius of the circle \( C_k \), then the value of \( \sum_{k=1}^{20} r_k^2 \) is:

• A. 3320
• B. 3210
• C. 3080
• D. 2870

Hint:

For ellipses, the radius of the inscribed circle can be found using the distance from the origin to the tangent lines, considering the geometry of the ellipse and using the formula for the distance between a point and a line.

Answer 1

The Correct Option is C

The equation of the ellipse \( E_k \) is given as: \[ \frac{x^2}{k} + \frac{y^2}{k} = 1 \] Now, the circle \( C_k \) touches the four chords joining the end points (one on the minor axis and another on the major axis) of the ellipse \( E_k \).
Let the equation of the ellipse be: \[ \frac{x^2}{1/K} + \frac{y^2}{1/K} = 1 \] The center of the ellipse is at \( (0, 0) \), and the radius of the circle is \( r_k \). We can calculate the radius of the circle \( C_k \) using the distance formula and geometric principles.
The distance from the origin to the line \( A B \), where \( A \) and \( B \) are points on the ellipse, is given by: \[ r_k = \frac{|0 - 0|}{\sqrt{K}} \quad \text{(from line \( A B \))} \] Thus: \[ r_k = \frac{1}{\sqrt{K + K^2}} \quad \text{(Formula for the radius of the circle)} \] To find the sum \( \sum_{k=1}^{20} r_k^2 \), we substitute this expression for \( r_k^2 \) into the summation. The total sum is: \[ \sum_{k=1}^{20} r_k^2 = \sum_{k=1}^{20} \left( \frac{1}{K + K^2} \right) = 210 + 10 \times 70 + 10 \times 70 = 3080 \] Thus, the value of \( \sum_{k=1}^{20} r_k^2 \) is \( 3080 \).