Question

Find the radius of the curvature Y= log sec x at any point

Answer 1

Answer:

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Answer 2

Given,

The given curve is Y= log sec x.

To find,

The radius of the curvature (ρ) for the given curve.

Solution,

The radius of curvature of Y= log sec x at any point is sec x.

We can simply find the radius of the curvature by using the formula

ρ = (1+y₁²)³/² / y₂                (1)

Y= log sec x

Differentiating Y w.r.t. x on both sides

Y₁ = 1/Sec x * Sec x tan x

Y₁ = tan x

Again differentiating Y₁  w.r.t. x on both sides, we get

Y₂ = sec²x

Substituting the values of Y₁ and Y₂, we get

ρ = (1+(tan²x)³/₂ / sec²x

ρ = (1+ tan²x)³/₂ / sec²x

Replacing sec²x with (1+tan²x)

ρ = (1+tan²x)³/₂ / (1+tan²x)

ρ = (1+tan²x)³/₂⁻¹      (aⁿ/aˣ = aⁿ⁻ˣ)

ρ = (1+tan²x)¹/₂

Replacing (1+tan²x) with sec²x

ρ = (sec²x)¹/₂

ρ = sec x

Hence, the radius of curvature of Y= log sec x at any point is sec x.