Find the length of the chord of the ellipse x²/25 + y²/16 = 1, whose middle point is (1/2, 2/5).

To find the length of the chord of the ellipse given by the equation x²/25 + y²/16 = 1, with its midpoint at (1/2, 2/5), we can follow these steps:

Step 1: Identify the Ellipse Parameters

The ellipse has a semi-major axis of a = 5 (since 25 = 5²) and a semi-minor axis of b = 4 (since 16 = 4²).

Step 2: Use the Chord Length Formula

The length of a chord of an ellipse can be calculated using the formula:

L = 2 * b * sqrt(1 - (h²/a²) - (k²/b²))

where (h, k) is the midpoint of the chord.

Step 3: Substitute the Midpoint Values

Here, h = 1/2 and k = 2/5. Now, we need to calculate:

• h² = (1/2)² = 1/4
• k² = (2/5)² = 4/25

Step 4: Calculate the Terms

Now, substitute these values into the formula:

• h²/a² = (1/4) / 25 = 1/100
• k²/b² = (4/25) / 16 = 1/100

Step 5: Find the Value Inside the Square Root

Now, calculate:

1 - (1/100) - (1/100) = 1 - 2/100 = 98/100 = 49/50

Step 6: Calculate the Chord Length

Now plug this back into the chord length formula:

L = 2 * 4 * sqrt(49/50)

L = 8 * (7/sqrt(50)) = 56/sqrt(50) = 56 * (sqrt(50)/50) = (56 * sqrt(50))/50

Final Result

The length of the chord is approximately 3.95 units.