The equation of the common tangent to the curves and is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
For the parabola y² = 8x, the tangent with slope m is y = mx + 2/m.
For the hyperbola xy = -1, substituting y = mx + 2/m: x(mx + 2/m) = -1 → mx² + (2/m)x + 1 = 0. For tangency, discriminant = 0: (2/m)² - 4m = 0 → 4/m² = 4m → m³ = 1 → m = 1.
So the common tangent is y = x + 2.
For the hyperbola xy = -1, substituting y = mx + 2/m: x(mx + 2/m) = -1 → mx² + (2/m)x + 1 = 0. For tangency, discriminant = 0: (2/m)² - 4m = 0 → 4/m² = 4m → m³ = 1 → m = 1.
So the common tangent is y = x + 2.
[AIEEE 2002]