Question

How to find the radius of the curvature of 16x^2 + 25y^2 = 400 at one end of the minor axis? MEDAL

Answer 1

This is an ellipse
I recommend changing it to standard form by dividing everything by the RHS.
Then you just find the semi minor axis.

Answer 2

@mww what is RHS?

Answer 3

RHS is right hand side
You want it to eventually look like this

\[\frac{ x^2 }{ a^2 }+\frac{ y^2 }{ b^2 }=1\]
which describes an ellipse at the origin with semi axes of a and b units.
I

Answer 4

@mww then how will i get the radius of the curvature?

Answer 5

well show me the equation you obtain first

Answer 6

When you arrive at the standard form this is what the ellipse looks like:



You can see how the 'a' and 'b' from standard form become the intercepts of your ellipse onto the xy axes. Thus your job is to find what your a and b are in the standard form

Answer 7

x^2/25 + y^2/16 =1

Answer 8

great of now the next step is easy. what is a and b?

Answer 9

a=5,-5 and b=4,-4

Answer 10

ok which gives you a semi major axis of 5 and semi minor axis of 4. (half of the length)

Answer 11

and then?

Answer 12

@mww what will happen next?

Answer 13

well 'radius' is technically one of those just think of squished radius of a circle. I'm a little confused as to what they mean by radius of curvature at minor end. I think it means this:

Answer 14

The better non ambiguous term would be semi-minor axis length.

Answer 15

@mww thank you

Answer 16

there's a load of stuff on Wiki about the Radius of Curvature if you're interested.

the simplest expression for it might be this [pinched straight from Wiki page - ish]

\(R = \dfrac{|\vec v|^3}{|\vec v \times \vec a|}\)

where:

\( \vec r = < 5 \cos t, 4 \sin t> \), ie the paramaterisation of your ellipse

So
\( \vec v = \frac{d}{dt}( \vec r) = (- 5 \sin t, 4 \cos t ) \)

and \( \vec a = \frac{d}{dt} (\vec v) = <-5 \cos t, - 4 \sin t> \)


usefully, \(|\vec v \times \vec a| = 20 \) for all \(t\)

and \(|\vec v| = 5\) for \(t = \pi /2\), which is the minor (y) axis

so \(R = 25/4\)

for the other axis, \(t = 0\) and \(|\vec v| = 4\)

so we get \(16/5\) there

this checks out with the quick formulae on one of the Wiki pages

working out what all of that means is really interesting :-)