Hello friends, today it’s all about the radius of curvature of any curve. Have a look!!

### Radius of curvature

Well, it’s an old topic from high school. So I’ll not go into much detail.

Suppose is the equation of any curve.

The equation of the radius of curvature at any point is

(1)

Now I will give you an example.

#### Example of the radius of curvature of any curve

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the example.

##### Example

According to Stroud and Booth (2013)* “If with prove that the radius of curvature at the point is ”

##### Solution

In this example, the equation of the curve is

Now, in order to know its radius of curvature, I have to start with the differentiation.

You can also check out the rules for differentiation

###### Step 1

First of all, I’ll differentiate the given equation with respect to .

Thus it will be

Here is a constant with a value greater than 0.

On the right hand side, is a product of two functions: and .

So, for that part, I’ll use the product rule of differentiation.

You can also check out the product rule of differentiation.

Thus it will be

Therefore I can say

(2)

As a next step, I’ll differentiate equation (2) with respect to to get the value of

###### Step 2

Now I’ll differentiate the given equation (2) with respect to .

So it will look like

Once more, I’ll use product rule of differentiation on both sides to get

Therefore I can say

(3)

Here I have to find out the radius of curvature at the point

For that, I’ll start with the value of at the point

###### Step 3

Now I’ll put and in equation (2).

Hence it will be

Next I’ll simplify it to get the value of at the point as

Thus I have

(4)

at the point .

Now I’ll get the value of at the point

###### Step 4

For that, I’ll substitute and in equation (3).

Thus it will look like

Next, I’ll simplify it to get the value of at the point as

So finally I get the value of at the point as

(5)

Now comes the last step of this problem.

###### Step 5

This is the last part of the problem.

Here I’ll substitute the values of and at the point from equations (4) and (5) to equation (1).

Thus the radius of curvature of the given curve at the point will be

Hence I have managed to prove that the radius of curvature of this curve at the point is

This is the answer to the given example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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