Question

If m be the slope of a tangent to the curve e^2y = 1 + 4x², then ______.

Options

• m < 1

• |m| ≤ 1

• |m| > 1

• |m| ≥ 1

Answer 1

If m be the slope of a tangent to the curve e^2y = 1 + 4x², then |m| ≤ 1.

Explanation:

e^2y = 1 + 4x²

2y = log_e(1 + 4x²)

y = `1/2log_e(1 + 4x^2)`

`(dy)/(dx) = 1/2 xx 1/(1 + 4x^2) xx 4 xx 2x = (4x)/(1 + 4x^2)`

`(dy)/(dx) = (4x)/(1 + 4x^2)` = m

⇒ 4mx² – 4x + m = 0

for x ∈ R,

Discriminant ≥ 0

⇒ 16 – 16 m² ≥ 0

⇒ |m| ≤ 1