Question

Three normals are drawn to the parabola y² = 4x from the point (c, 0). These normals are real and distinct when –

• A. c = 0
• B. c = 1
• C. c = 2
• D. c = 3

Answer 1

Answer: (D)

c = 3

Answer 2

Any point on y² = 4x is (t², 2t). Since $\frac{dy}{dx}$ = $\frac{2}{y}$ so $\left. \ \frac{dy}{dx} \right|_{(t^{2},2t)}$

= $\frac{2}{2t}$ = $\frac{1}{t}$. Hence equation of normal at (t², t) is

Y – 2t = –t (X – t²)

This passes through (c, 0) if –2t = – t (c – t²)

̃ t = 0, t² = c – 2

Thus the roots are real and distinct if c > 2

so c = 3 is the correct choice.