If y = mx + 4 is a tangent to both the parabolas, y² = 4x and x² = 2by, then b is equal to :

Solution

Maths sir · 4 steps

Get the slope from the first parabola, then make the line touch the second one exactly once.

1. Slope from first parabola. For y² = 4ax, the line y = mx + c is a tangent when c = a/m. Here a = 1 and c = 4, so 1/m = 4, which gives m = 1/4.
2. Meet the second parabola. Put y = x/4 + 4 into x² = 2by. This gives x² = b/2x + 8b, that is, x² - b/2x - 8b = 0.
3. Touch means one root. A tangent meets the parabola at exactly one point, so the discriminant of this quadratic must be zero.

   (b/2)² + 32b = 0 ⇒ b(b/4 + 32) = 0

4. So. b = 0 does not give a parabola, so b/4 = -32 and b = -128. The answer is (a).

The idea to keep. For a common tangent, use the ready-made tangent condition on the simpler parabola to get the slope. Then substitute into the other parabola and set the discriminant to zero.

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