# 1. Tangents and Normals

by M. Bourne

We often need to find tangents and normals to curves when we are analysing forces acting on a moving body.

A **tangent** to a curve is a line that touches the curve at one point and has the same **slope** as the curve at that point.

A **normal** to a curve is a line **perpendicular** to a tangent to the curve.

the curve

the curve

Graph showing the tangent and the normal to a curve at a point.

**Note 1:** As we
discussed before (in Slope of a Tangent to a Curve), we can find the slope of a tangent at any point (*x*, *y*) using `dy/dx`.

**Note 2: **To find the equation of a normal, recall the
condition for two lines with slopes *m*_{1}and* **m*_{2} to be perpendicular (see Perpendicular Lines):

m_{1}×m_{2}= −1

### Applications

A car has skidded while rounding a corner, **tangent** to the double yellow lines curve.

**Tangent: **

- If we are traveling in a car around a corner and we drive over something slippery on the road (like oil, ice, water or loose gravel) and our car starts to skid, it will continue in a direction
**tangent**to the curve. - Likewise, if we hold a ball and swing it around in a circular motion then let go, it will fly off in a
**tangent**to the circle of motion.

The spokes of a bicycle wheel are **normal** to the rim.

**Normal: **

- When you are going fast around a circular track in a car, the force that you feel pushing you outwards is
**normal**to the curve of the road. Interestingly, the force that is making you go around that corner is actually directed towards the**center**of the circle, normal to the circle. - The spokes of a wheel are placed
**normal**to the circular shape of the wheel at each point where the spoke connects with the center.

### Examples

### Need Graph Paper?

**1. **Find the **gradient** of

(i) the tangent (ii) the normal

to the curve *y*
= *x*^{3}
− 2*x*^{2} + 5 at the point `(2,5)`.

Answer

`dy/dx=3x^2-4x`

The slope of the **tangent** is

`m_1=[dy/dx] _(x=2)`

`=3(2)^2-4(2)`

`=12-8`

`=4`

The slope of the **normal** is found using *m*_{1} ×
*m*_{2} = −1

`m_2=-1/4`

**2. **Find the **equation** of (i) the tangent and (ii) the normal in the above
example.

Answer

We use *y* − *y*_{1} = *m*(*x* − *x*_{1}), with *x*_{1} = 2, *y*_{1} = 5

(i) The tangent has slope `4`, so we have:

`y-5=4(x-2)`

gives

`y=4x-3`

or

4

x−y− 3 = 0

(ii) Now for the normal to the curve. Since the tangent has slope `4`, we have the slope of the normal `m=-1/4`

So we substitute as follows:

`y-5=-1/4(x-2)`

gives

`y=-1/4x+5 1/2`

or

x+ 4y− 22 = 0

**3. ****Sketch** the curve and
the normal in the above example.

Answer

Here's the graph of the tangent and normal to the curve at `x=2`.