Hello. This lesson will talk about various equations of a normal to a circle.

As mentioned earlier, this won’t be much complicated and all equations will be based on one simple fact – ‘the normal to a circle always passes through its center’. Let’s begin.

## Equation of a normal to the circle x^{2} + y^{2} = a^{2} at a given point (x_{1}, y_{1})

Here’s what the situation looks like:

The given normal passes through the point (x_{1}, y_{1}) and will also pass through the center of the circle, i.e (0, 0). Now, to find the equation of the normal, all we have to do is use the two-point form of the equation of a straight line.

The required equation will be (y – 0)/(y_{1} – 0) = (x – 0)/(x_{1} – 0) or **y/y _{1} = x/x_{1}**. That’s it!

Pretty simple, as I told. Let’s move on to the other cases.

## Equation of a normal to the circle x^{2} + y^{2} = a^{2} from a given point (x_{1}, y_{1})

In this case, the given normal will again pass through the point (x_{1}, y_{1}) and the center of the circle, except that the point (x_{1}, y_{1}) does not lie on the circle.

We’ll use the the two-point form again – the required equation will be (y – 0)/(y_{1} – 0) = (x – 0)/(x_{1} – 0) or **y/y _{1} = x/x_{1}**.

Moving on…

## Equation of a normal, of given slope ‘m’, to the circle x^{2} + y^{2} = a^{2}

In this case, the normal will still pass through the center. Additionally, we are given it’s slope. So we’ll be using the point-slope form to find its equation.

The required equation will be y – 0 = m(x – 0), or **y = mx**

The remaining three cases are similar to the previous three, except that the center of the circle will now be (-g, -f), instead of (0, 0).

## Equation of a normal to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 at a given point (x_{1}, y_{1})

Similar to case (1) above, the equation of the normal, using the two-point form, will be **(y – y _{1})/(y_{1} + f) = (x – x_{1})/(x_{1} + g)**

## Equation of a normal to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 from a given point (x_{1}, y_{1})

This one is similar to case (2) above. The equation of the normal in this case will be again **(y – y _{1})/(y_{1} + f) = (x – x_{1})/(x_{1} + g)**

## Equation of a normal, of given slope ‘m’, to the circle x^{2} + y^{2} + 2gx + 2fy + c = 0

Lastly, similar to case (3), using the point-slope form, the equation of the normal will be **(y + f) = m(x + g)**

That pretty much covers everything about equations to the normals. The next lesson will cover a few examples, and also one very important application of normals – finding the shortest distance between two curves.

See you there!