Hello friends, today it’s all about the tangent, normal to any curve. Have a look!!

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**Tangent, normal to any curve**

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 tangent of this curve at the point is

Similarly, the equation of the normal to this curve at the point is

Now I will solve some problems for you.

**Examples on the tangent, normal to any curve**

Disclaimer: 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 first example.

##### Example 1

According to Stroud and Booth (2013)* “Find the equations of the tangent and normal to the curve at the point (6, 4).”

##### Solution

Now here the given curve is

And I have to find out the equations of the tangent and normal to this curve at the point (6, 4). So I’ll start with the value of at this point.

###### Step 1

First of all, I’ll differentiate both sides of the given curve with respect to . and that gives

Next, I’ll simplify it to get

Then I’ll cancel from both sides to get

So I can say the value of is

Thus the value of at the point (6, 4) will be

Now I’ll simplify it to get

And here comes my next step.

**Step 2**

As per the formula, the equation of the tangent to the curve at the point (6, 4) is

Now I will simplify it further. So that gives

Next, I’ll do some simple aritmetic calculation like

And this is the equation of the tangent. Now I’ll get the equation of the normal.

**Step 3**

As per the formula, the equation of the normal to the curve at the point (6, 4) is

Next, I will simplify it further to get

Then I’ll do some simple aritmetic calculation like

And this is the equation of the normal.

Hence I can conclude that the equations of the tangent and normal to the curve at the point (6, 4) are and respectively.

And this is the answer to the given example. Now I will give another example.

##### Example 2

According to Stroud and Booth (2013)* “If find the equation of the tangent at ”

##### Solution

Now here the given parametric curves are

(1)

and

(2)

And I have to find out the equation of the tangent at . First of all, I’ll find out the value of at

**Step 1**

Since both equations (1) and (2) are parametric curves, I’ll do it in the same way as the * differentiation of parametric curves*.

So, first I’ll differentiate both sides of equation (1) with respect to to get

Next, I’ll differentiate both sides of equation (2) with respect to to get

Thus, as per the rule, will be

So here it will be

Now at will be

Next, I’ll simplify it to get

which in turn gives

So that means

Now I’ll get the values of and at

**Step 2**

First, I’ll substitute in the equation (1) to get the value of at . Thus it will be

As I know

So the value of at will be

Next, I’ll substitute in the equation (2) to get the value of at . Thus it will be

Again will be

So the value of at will be

Therefore I can say that for , the point will be

**Step 3**

Now I’ll get the equation of the tangent at the point Thus it will be

Next, I’ll simplify it to get

Then I’ll do some simple arithmetic to get the value of as

Thus I can say that the equation of the tangent at is

And this is the answer to the given example.

Dear friends, this is the end of 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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