Question

The set of points on the axis of the parabola y² = 4x + 8 from which the three normals to the parabola are all real and different is –

• A. {(k, 0) | k £ –2}
• B. {(k, 0) | k > – 2}
• C. {(0, k) | k > –2}
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

The equation of the normal at (–2 + t², 2t) is

y – 2t = $\frac{- 1}{\left( \frac{dy}{dx} \right)_{( - 2 + t^{2},2t)}}$· (x + 2 – t²) is

Ž y – 2t = $–\frac{2t}{2}$. (x + 2 – t²)

Ž tx + y = 2t – 2t + t³

Ž tx + y = t³

It passes through (k, 0), if tk = t³.

Ž t(t² – k) = 0.

It has three real values of t, if k > 0. So, the set of points (k, 0) is such that k > 0.