Question

Find radius of curvature of curve: x 2 + xy + y 2 = 4 at point (–2, 0)

Answer 1

Given :  curve x² + xy + y² = 4

To find : radius of curvature of curve: at radius of curvature of curve:

Solution:

x² + xy + y² = 4

2x + x(dy/dx) + y  + 2y(dy/dx)  = 0

=> (dy/dx)(x + 2y) =  -(2x + y)

=> dy/dx =  -(2x + y)/(x + 2y)

x = - 2  , y = 0

=> dy/dx =  -(-4 + 0)/(-2 + 0)  =  - 2

2x + x(dy/dx) + y  + 2y(dy/dx)  = 0

=> 2 + x(d²y/dx²)  + dy/dx  + 2y((d²y/dx²) + 2(dy/dx)(dy/dx) = 0

putting x = - 2  , y = 0 & dy/dx = - 2

=> 2 -2(d²y/dx²)  - 2  + 0  + 2(-2)(-2) = 0

=> d²y/dx² = 4

R =  (  √ (1  + (dy/dx)²)³ )/ (d²y/dx²)

=> R = √(( 1  + (-2)²)³) / (4)

=> R =   √125  / 4

Radius of curvature of curve: =   √125  / 4

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