Question

The length of the chord of the ellipse \(\frac{x^2}{25} + \frac{y^2}{16} = 1\), whose mid-point is \(\left(1, \frac{2}{5}\right)\), is equal to:

• A. \(\frac{\sqrt{1691}}{5}\)
• B. \(\frac{\sqrt{2009}}{5}\)
• C. \(\frac{\sqrt{1741}}{5}\)
• D. \(\frac{\sqrt{1541}}{5}\)

Answer 1

The Correct Option is A

Given the ellipse:

\(\frac{x^2}{25} + \frac{y^2}{16} = 1\) and a chord with midpoint \(\left( 1, \frac{25}{8} \right)\).

Step 1. Equation of the Chord: The chord equation is:
\(\frac{x}{25} + \frac{y}{40} = 1 \Rightarrow y = \frac{200 - 8x}{5}\)
 

Step 2. Substitute into the Ellipse: Substituting \( y \) gives:

\(\frac{x^2}{25} + \frac{\left( \frac{200 - 8x}{5} \right)^2}{16} = 1\)
 

 Simplifying:

  \(2x^2 - 80x + 990 = 0 \Rightarrow x = 20 \pm \sqrt{10}\)
 

Step 3. Length of the Chord: Using the distance formula, the length is:
 \(\text{Length} = \frac{\sqrt{1691}}{5}\)