The length of the chord of the ellipse x²/25+y²/16=1, whose mid point is (1, 2/5), is equal to :

Solution

Maths sir · 4 steps

Midpoint given? Write the chord with T = S₁, then use the sum and product of roots to get its length.

1. Chord from midpoint. For a chord of x²/a²+y²/b²=1 with midpoint (x₁, y₁), the equation is T = S₁, that is xx₁/a²+yy₁/b² = x₁²/a²+y₁²/b².

   x/25+y/40 = 1/25+1/100 = 1/20 ⇒ 8x + 5y = 10

2. Meet the ellipse. Put y = 2 - 8x/5 into 16x² + 25y² = 400. This gives 80x² - 160x - 300 = 0, or 4x² - 8x - 15 = 0. As a check, x₁ + x₂ = 2, so the midpoint has x = 1 as it should.
3. Gap in x. (x₁ - x₂)² = (x₁+x₂)² - 4x₁x₂ = 4 + 15 = 19. The slope is -8/5, so (y₁ - y₂)² = 64/25 × 19.
4. So. Length² = 19(1 + 64/25) = (19 × 89)/25 = 1691/25, so the length is √1691/5, option (a).

The idea to keep. For a chord with a known midpoint, T = S₁ gives its equation at once. For its length, do not solve for the endpoints: use (x₁ - x₂)² from Vieta and multiply by 1 + m².

Answer from the NTA final answer key. Solution by PrepWiser.