Normal: Equations

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 x2 + y2 = a2 at a given point (x1, y1)

Here’s what the situation looks like:

The given normal passes through the point (x1, y) 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)/(y­1 – 0) = (x – 0)/(­x1 – 0) or y/y1 = x/x1. That’s it!

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

Equation of a normal to the circle x2 + y2 = a2 from a given point (x1, y1)

In this case, the given normal will again pass through the point (x1, y) and the center of the circle, except that the point (x­1, y1) does not lie on the circle.

We’ll use the the two-point form again – the required equation will be (y – 0)/(y­1 – 0) = (x – 0)/(­x1 – 0) or y/y1 = x/x1.

Moving on…

Equation of a normal, of given slope ‘m’, to the circle x2 + y2 = a2

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 x2 + y2 + 2gx + 2fy + c = 0 at a given point (x1, y1)

Similar to case (1) above, the equation of the normal, using the two-point form, will be (y – y­­1)/(y1 + f) = (x – x1)/(x1 + g)

Equation of a normal to the circle x2 + y2 + 2gx + 2fy + c = 0 from a given point (x1, y1)

This one is similar to case (2) above. The equation of the normal in this case will be again (y – y­­1)/(y1 + f) = (x – x1)/(x1 + g)

Equation of a normal, of given slope ‘m’, to the circle x2 + y2 + 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!