Question

show that the radius of the curvature at point (r, theta) of the curve r^2 cos2theta = a^2 is r^2/a^2​

Answer 1

Answer:

Write the derivatives of the quadratic function:

y

′

=

(

x

2

)

′

=

2

x

;

y

′

′

=

(

2

x

)

′

=

2.

Then the curvature of the parabola is defined by the following formula:

K

=

y

′

′

[

1

+

(

y

′

)

2

]

3

2

=

2

[

1

+

(

2

x

)

2

]

3

2

=

2

(

1

+

4

x

2

)

3

2

.

At the origin (at

x

=

0

), the curvature and radius of curvature, respectively, are

K

(

x

=

0

)

=

2

(

1

+

4

⋅

0

2

)

3

2

=

2

,

R

=

1

K

=

1

2

.

Example 3.

Find the curvature and radius of curvature of the curve

y

=

cos

m

x

at a maximum point.

Solution.

This function reaches a maximum at the points

x

=

2

π

n

m

,

n

∈

Z

.

By the periodicity, the curvature at all maximum points is the same, so it is sufficient to consider only the point

x

=

0.