Question

find thae radius of curvature at origin of the cutve ,y-x=x^2+2xy+y^2.
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Answer 1

Radius of the curvature is 4 units

Given:

Equation of the curvature y - x = x²+ 2xy + y²    

To find:

The radius of curvature at the origin

Solution:

Formula used:

Radius of curvature, $R = \frac{ [ 1 + (\frac{dy}{dx})^{2} ]^{ \frac{3}{2} } }{(\frac{d^{2} y }{dx^{2} })}$  

Here we have

Equation of the curvature is  y - x = x²+ 2xy + y²    

=> y = x²+ 2xy + y² - x

Differentiate the above equation with respect to x

=> $\frac{dy}{dx} = 2x + 2y + 0 + 1$

=> $\frac{dy}{dx} = 2x + 2y + 1$ ----(1)

Differentiate (1) with respect to x again

=> $\frac{d^{2} y }{dx^{2} } = 2 + 0 + 0$  

=> $\frac{d^{2} y }{dx^{2} } = 2$  ---- (2)

Substitute (1) and (2) in the given formula

=> Radius of curvature, $R = \frac{ [ 1 + (2x + 2y + 1)^{2} ]^{ \frac{3}{2} } }{(2)}$

Here given point is Origin i.e (0, 0)  

=> $R = \frac{ [ 1 + (2(0)+ 2(0) + 1)^{2} ]^{ \frac{3}{2} } }{(2)}$  

=> $R = \frac{ [ 2^{2} ]^{ \frac{3}{2} } }{(2)}$  

=>  $R = \frac{ [ 2^{3} ] }{(2)}$

=>  R = 4  

Therefore,

Radius of the curvature is 4 units  

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Answer 2

Answer: The answer to the above given question is 4.

Step-by-step explanation: The radius of curvature

R = (1+(dy / dx)²)3/2 / |d²y / dx²|

Given equation of curvature is

y - x = x² + 2xy + y²

This equation can be written as

y = x² + 2xy + y² + x

Differentiate above equation with respect to x

dy / dx = 2x + 2y + 0 + 1

dy / dx = 2x + 2y + 1 .........(1)

Differentiate equation 1 with respect to x

d²y / dx² = 2 + 0 + 0

d²y / dx² = 2 ...........(2)

Substitute equation 1 & 2 in radius of curvature formula

R = (1+(2x + 2y + 1)²)3/2 / |2|

Here given to find radius of curvature at origin.

So ( x , y ) = ( 0 , 0 )

Substitute x & y values in above equation.

R = (1+(2(0) + 2(0) + 1)²)3/2 / |2|

  = ((1 +1)²)3/2 / |2|.      ( 1 can also be written as (1)²

R = ((2)²)3/2 / |2|.

R = (2)3 / |2|.

R = 4

So the radius of curvature is 4 units.

Here are two links provide below which helps to know about relationship between curvature and radius of curvature and what is centre of curvature and radius of curvature.

https://brainly.in/question/10242372

https://brainly.in/question/1181551

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