Question

The length of focal chord of the hyperbola xy = 2 which touches the circle x2+y2 = 2 is

Answer 1

The length of focal chord of the hyperbola is 16

Step-by-step explanation:

The slope of the focal chord is given by the formula:

mAB = -1/(t₁t₂) = (2√2/t₁ - 4)/(2√2t₁ - 4)

- 2√2t₁ + 4 = 2√2t₂ - 4t₁t₂

4 + 4t₁t₂ = 2√2t₂ + 2√2t₁

4 (1 + t₁t₂)  = 2√2 (t₂ + t₁) → (equation 1)

Now,

AB = √((2√2t₁ - 2√2t₂)² + (2√2/t₁ - 2√2/t₂)²)

AB = 2√2 |t₁ - t₂|/|t₁t₂| √(t₁²t₂² + 1) → (equation 2)

|t₁ - t₂| = √((t₁ + t₂)² - 4t₁t₂)

On using equation (1), we get,

|t₁ - t₂| = √(2(1 + t₁t₂)² - 4t₁t₂) = √2 √(1 + t₁²t₂²)

On using equation (2), we get,

AB = (4 (1 + t₁²t₂²))/(|t₁t₂|)

AB = 4/(sin π/12 × cos π/12)

∴ AB = 16