Consider the ellipse 9 x^2+y^2=36. (a) Compute the radius of curvature in terms of x and y. (b)

Consider the ellipse 9 x^2+y^2=36. (a) Compute the radius of...

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of points whose distances to two fixed points (foci) have a constant sum. In analytical geometry, ellipses are described by quadratic equations in x and y, and their study involves understanding properties such as axes lengths, symmetry, and curvature at various points.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is given in an implicit form. Rather than solving the equation for one variable in terms of the other, differentiation is applied directly to both sides of the equation, allowing one to compute the slope of the tangent to the curve even when the relationship between the variables is not expressed as an explicit function.

Curvature and Radius of Curvature

Curvature measures how sharply a curve bends at a given point. The radius of curvature is the reciprocal of curvature and represents the radius of the osculating circle—the circle that best approximates the curve near that point. These concepts are key in differential geometry and are used to analyze the local behavior of curves in a variety of contexts.

Transcript

00:01 I want to find the radius of curvature for this ellipse at these three points, and then look at the picture.

00:08 All right, so first we need the first derivative for here, and the second derivative for here.

00:15 So 18x plus 2yy prime equals 0.

00:22 So y prime is minus 18x over 2y, so minus 9x over y.

00:29 I'm going to take the derivative of both sides and find the second derivative.

00:36 So y double prime, the bottom, derivative of the top, minus the top times the derivative of the bottom, over the bottom squared.

00:50 So minus 9y plus 9xy over y squared.

00:58 Now in place of this y prime, i'm going to put this.

01:02 So i have minus 9y plus 9x times what was it? i moved it too fast.

01:13 Minus 9x over y over y squared.

01:24 Okay, so now i'm going to put these all over 1, and i'm going to multiply everything by y to get rid of this fraction right here.

01:36 So now i have minus 9y squared, minus 81x squared, over y cubed.

01:49 So that's negative 9 times 9x squared plus y squared over y cubed, but 9x squared plus y squared is equal to 36 from the beginning.

02:07 So i'm going to put that in.

02:10 So now i have negative 9 times 36 over y cubed.

02:22 For 3 minus 324 maybe over y cubed.

02:29 All right, so now we're gonna put that in here.

02:34 So r is 1 plus minus minus minus 9x over y squared to the three halves over the absolute value of the second derivative, which will be 324 over y cubed.

02:55 So that's 1 plus 81 x squared over y squared to the three halves over 324 over y cubed.

03:11 Okay, so get a common denominator inside those parentheses.

03:14 So you have y squared plus 81 x squared to the three halves over y squared to the three halves over 324 over y cubed so now you have y squared plus 81 x squared to the three halves over y to the three halves times y to the third oh no why to the why to the third down there because it's y squared to the three halves so why to the third i'm flipping here so that i can just cancel those okay so r is y squared plus 81 x squared to the three halves over 324.

04:11 No, you can't bring that in there and make this y cubed an 81 to the three halves x to the third.

04:18 Okay, for the same reason you can't do this because it's just not true.

04:26 All right, so don't bring that in there.

04:29 All right, so the points are negative to zero.

04:32 So when we plug that one in, we get r equals 0 plus 81 times 4 to the three halves over 3 .24.

04:44 Okay, the square root of 81, 9 cubed.

04:47 I don't know, i'm gonna put it like this.

04:50 Square of 4, 2, 2 cubed is 8...