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

Answer: (A)

$\frac{\sqrt{1691}}{5}$

Answer 2

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:
$\text{Length} = \frac{\sqrt{1691}}{5}$