Question

If the three normals drawn to the parabola, y² = 2x pass through the point (a, 0)a ≠ 0, then' a' must be greater than ______.

Options

• 1

• `1/2`

• `-1/2`

• –1

Answer 1

If the three normals drawn to the parabola, y² = 2x pass through the point (a, 0)a ≠ 0, then' a' must be greater than 1.

Explanation:

Given, parabola is y² = 2x  ...(i)

Let the equation of the normal is y = mx – 2am – am³  ...(ii)

Using equation (i)

4a = 2  ...(Standard Equation of parabola y² = 4ax)

⇒ a = `1/2`  ...(iii)

Using Equation (ii) and (iii)

y = `mx - m - 1/2m^3`

Given that normal passes through the point (a, 0)

Hence, 0 = `m(a) - m - 1/2(m)^3`

⇒ `m(a - 1 - m^2/2)` = 0

⇒ m = 0 or `a - 1 - m^2/2` = 0

⇒ m = 0 or m² = 2(a – 1)

As m² > 0 ⇒ 2(a – 1) > 0

a – 1 > 0

a > 1

Hence, 'a' must be greater than one.