To find the radius of curvature of the curve given in the form y=y(x0) at some point (x0,y(x0))=(x0,y0) we can use the following formula
R(x0)=∣y′′(x0)∣[1+(y′(x0))2]3/2
To find the derivatives we can use the method of implicit differentiation. Let's write the equation of the curve
x3+y3−3axy=0
and differentiate both halves
3x2+3y2y′−3ay−3axy′=0group the terms
3(y2−ax)y′+3x2−3ay=0
and expressing y′
y′=3(y2−ax)3ay−3x2=y2−axay−x2Remembering this result return to the expression
3x2+3y2y′−3ay−3axy′=0 Differentiate it once again
6x+6y(y′)2+3y2y′′−3ay′−3ay′−3axy′′=0 group the terms
3(y2−ax)y′′+6(x+y(y′)2−ay′)=0
y′′=2y2−axay′−y(y′)2−xNow we can do all calculations using x=x0=3a and y=y0=3a . Let's begin with the first derivative.
y′(x0)=y02−ax0ay0−x02=9a2−3a23a2−9a2=−1 And with the second derivative, using the value of the first
y′′(x0)=2y02−ax0ay′(x0)−y0(y′(x0))2−x0=29a2−3a2−a−3a−3a=−26a27a=−3a7 At least we can use the formula for the radius of curvature
R(x0)=∣y′′(x0)∣[1+(y′(x0))2]3/2=3∣a∣7(1+1)3/2=73∣a∣⋅23/2=2273∣a∣=762∣a∣ This is the answer
Comments