Question

The equation to the line touching both the parabolas y² = 4x and x² = –32y is ______.

Options

• x + 2y + 4 = 0

• 2x + y – 4 = 0

• x – 2y – 4 = 0

• x – 2y + 4 = 0

Answer 1

The equation to the line touching both the parabolas y² = 4x and x² = –32y is x – 2y + 4 = 0.

Explanation:

Let equation of tangent to y² = 4x is y = `mx + 1/m`

This tangent also touches x² = –32y

∴ x² = `-32(mx + 1/m)`

mx² = –32(m²x + 1)

⇒ mx² + 32m²x + 32 = 0

For equal roots,

D = 0

(32 m²)² – 4.m.32 = 0

⇒ 32m² = 4  ...(∵ m ≠ 0)

⇒ m² = `1/8`

⇒ m = `1/2`

∴ Equation of line is y = `1/2.x + 1/(1/2)`

⇒ y = `x/2 + 2`

⇒ x – 2y + 4 = 0