Question #95117
Find the radius of curvature of follium decart x^3+y^3=3axy at the point (3a,3a)
1
Expert's answer
2019-09-24T11:35:24-0400

To find the radius of curvature of the curve given in the form y=y(x0)y = y({x_0}) at some point (x0,y(x0))=(x0,y0)({x_0},y({x_0})) = ({x_0},{y_0}) we can use the following formula


R(x0)=[1+(y(x0))2]3/2y(x0)R({x_0}) = \frac{{{{[1 + {{(y'({x_0}))}^2}]}^{3/2}}}}{{\left| {y''({x_0})} \right|}}


To find the derivatives we can use the method of implicit differentiation. Let's write the equation of the curve


x3+y33axy=0{x^3} + {y^3} - 3axy = 0


and differentiate both halves


3x2+3y2y3ay3axy=03{x^2} + 3{y^2}y' - 3ay - 3axy' = 0

group the terms


3(y2ax)y+3x23ay=03({y^2} - ax)y' + 3{x^2} - 3ay = 0


and expressing yy'


y=3ay3x23(y2ax)=ayx2y2axy' = \frac{{3ay - 3{x^2}}}{{3({y^2} - ax)}} = \frac{{ay - {x^2}}}{{{y^2} - ax}}

Remembering this result return to the expression


3x2+3y2y3ay3axy=03{x^2} + 3{y^2}y' - 3ay - 3axy' = 0

Differentiate it once again


6x+6y(y)2+3y2y3ay3ay3axy=06x + 6y{(y')^2} + 3{y^2}y'' - 3ay' - 3ay' - 3axy'' = 0

group the terms


3(y2ax)y+6(x+y(y)2ay)=03({y^2} - ax)y'' + 6(x + y{(y')^2} - ay') = 0

y=2ayy(y)2xy2axy'' = 2\frac{{ay' - y{{(y')}^2} - x}}{{{y^2} - ax}}

Now we can do all calculations using x=x0=3ax = {x_0} = 3a and y=y0=3ay = {y_0} = 3a . Let's begin with the first derivative.


y(x0)=ay0x02y02ax0=3a29a29a23a2=1y'({x_0}) = \frac{{a{y_0} - x_0^2}}{{y_0^2 - a{x_0}}} = \frac{{3{a^2} - 9{a^2}}}{{9{a^2} - 3{a^2}}} = - 1

And with the second derivative, using the value of the first


y(x0)=2ay(x0)y0(y(x0))2x0y02ax0=2a3a3a9a23a2=27a6a2=73ay''({x_0}) = 2\frac{{ay'({x_0}) - {y_0}{{(y'({x_0}))}^2} - {x_0}}}{{y_0^2 - a{x_0}}} = 2\frac{{ - a - 3a - 3a}}{{9{a^2} - 3{a^2}}} = - 2\frac{{7a}}{{6{a^2}}} = - \frac{7}{{3a}}

At least we can use the formula for the radius of curvature


R(x0)=[1+(y(x0))2]3/2y(x0)=(1+1)3/273a=3a723/2=223a7=627aR({x_0}) = \frac{{{{[1 + {{(y'({x_0}))}^2}]}^{3/2}}}}{{\left| {y''({x_0})} \right|}} = \frac{{{{(1 + 1)}^{3/2}}}}{{\frac{7}{{3\left| a \right|}}}} = \frac{{3\left| a \right|}}{7} \cdot {2^{3/2}} = 2\sqrt 2 \frac{{3\left| a \right|}}{7} = \frac{{6\sqrt 2 }}{7}\left| a \right|

This is the answer



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