Question

The equation of the line touching both the parabolas y² = x and x² = y is ______.

पर्याय

• 4x + 4y + 1 = 0

• 4x + 4y – 1 = 0

• x + y + 1 = 0

• 4x – 4y + 1 = 0

Answer 1

The equation of the line touching both the parabolas y² = x and x² = y is 4x + 4y + 1 = 0.

Explanation:

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

y = `mx + 1/(4m)`  ...(i)

This is also touches x² = y  ...(ii)

∴ From (1) and (2)

x² = `mx + 1/(4m)`

⇒ `x^2 - mx - 1/(4m)` = 0

D = 0

`m^2 + 4. 1/(4m)` = 0

m³ = –1

m = –1

∴ Required equation y = `-x - 1/4`

⇒ 4y = –4x – 1

⇒ 4x + 4y + 1 = 0