find radius of curvautre x₂y=a(x²+y²) at (-2a,2a)
Answers
Step-by-step explanation:
This curve is somewhat well-known; it is called the folium of Descartes.
There might be a sleeker way to do this, but I find that the usual curvature formula for plane curves works nicely enough: to find the curvature of the curve y=f(x) at the point (x0,f(x0)) , we have
κ(x0)=|f′′(x0)|[1+f′(x0)2]3/2
Even if our curve is not exactly a one-to-one function, locally, we can say it is, and implicit differentiation will save us here. Let’s pretend that y=f(x) for some f around the point (3a/2,3a/2) . Then, by differentiating both sides of the equation with respect to x, we get
3x2+3y2y′=3ay+3axy′
and isolating y’ yields
y′=3x2−3ay3ax−3y2=x2−ayax−y2
Then, to get y′′ , we just differentiate y′ with respect to :
(Exercise: show that .)
Substituting gives us , and so
The radius of curvature is, of course, .
You can also do it using a parametrization and the usual curvature formulas for parametric space curves, but I’m not too thrilled about all those quotient rules. Then again, all things considered, it might be the same amount of work. For your information, the usual parametrization is
and the desired point is given by .
What is the radius of curvature for the curve x³+y³=3axy at the point (3a/2, 3a/2)?
What is the radius of the curvature of x^2y=a (x^2+y^2) at (-2a,-2a)?
What is the solution of this question? Find the radius of curvature of the curve x^3+y^3=3xy at the point (3/2,3/2).
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What is the radius of curvature to the curve x to the power 2/3 +y to the power 2/3 = a to the power 2/3?
Answer:
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