Finding equation of osculating circles of ellipse

In summary: At (2,0), you can use the unit tangent vector and normal vector to find the center and radius of the osculating circle. In summary, to find the equations of the osculating circles of the ellipse 9x^2 + 4y^2 =36 at the points (2,0) and (0,3), you can use the concept of curvature and the unit tangent and normal vectors to calculate the center and radius of the osculating circles.
  • #1
digipony
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Homework Statement


Find the equations of the osculating circles of the ellipse 9x^2 + 4y^2 =36 at the points (2,0) and (0,3)


Homework Equations





The Attempt at a Solution


I honestly have no idea what to do here. This problem is in the chapter relating to curvature and arc length, as well as unit tangent/normal/binormal vectors for vector functions, but I do not see the relation of these topics to this problem. For some reason I have the idea of parametric equations come to mind. Could someone please point me in the right direction?
 
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  • #2
digipony said:

Homework Statement


Find the equations of the osculating circles of the ellipse 9x^2 + 4y^2 =36 at the points (2,0) and (0,3)


Homework Equations





The Attempt at a Solution


I honestly have no idea what to do here. This problem is in the chapter relating to curvature and arc length, as well as unit tangent/normal/binormal vectors for vector functions, but I do not see the relation of these topics to this problem. For some reason I have the idea of parametric equations come to mind. Could someone please point me in the right direction?

Do you know how to calculate the curvature at (0,3)? The osculating circle has the same radius as the radius of curvature of the ellipse and is tangent to the ellipse on the concave side.
 

FAQ: Finding equation of osculating circles of ellipse

What is the equation for an osculating circle of an ellipse?

The equation for the osculating circle of an ellipse is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.

How do you find the center and radius of an osculating circle of an ellipse?

The center of the osculating circle can be found by taking the average of the foci of the ellipse. The radius can be found using the formula r = a^2/b, where a is the semi-major axis of the ellipse and b is the semi-minor axis.

What is the significance of the osculating circle of an ellipse?

The osculating circle of an ellipse is the circle that best approximates the shape of the ellipse at a specific point. It is used in calculus to find the curvature of the ellipse at that point.

How is the equation of the osculating circle related to the equation of the ellipse?

The equation of the osculating circle is derived from the equation of the ellipse. It uses the same center point and radius, but with a different variable for the radius (r instead of a or b).

Can the osculating circle of an ellipse have a negative radius?

No, the radius of the osculating circle cannot be negative. It represents a distance and therefore must be positive. If the radius calculated is negative, it means there is an error in the calculation or the point chosen is not on the ellipse.

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