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