Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x² - y² = 3. If L is also a tangent to the parabola y² = αx, then α is equal to :

Solution

Maths sir · 5 steps

Use the hyperbola to fix k, then apply the parabola's tangent condition with the slope m = -2.

1. Slope form. Write L as y = -2x + k. So the slope is m = -2 and the intercept is c = k.
2. Tangent to hyperbola. For x²/a² - y²/b² = 1, the line y = mx + c is a tangent when c² = a²m² - b². Here a² = b² = 3, so k² = 3 × 4 - 3 = 9. Since k > 0, k = 3 and L is y = -2x + 3.
3. Tangent to parabola. Write y² = α x as y² = 4Ax with A = α/4. The line y = mx + c touches it when c = A/m.

   3 = (α/4)/-2 ⇒ α = -24

4. Check. Put x = (3 - y)/2 into y² = α x. This gives 2y² + α y - 3α = 0. A tangent needs a repeated root, so α² + 24α = 0. Since α ≠ 0, α = -24.
5. So. α = -24, which is option (d).

The idea to keep. For common tangents, put the line in y = mx + c form first. Then use c² = a²m² - b² for the hyperbola and c = a/m for the parabola, and keep the sign of m.

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