Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

Finding the equation of a circle

Hello again,

The question I'm having a little trouble with is this:

"2. Find the equation of the circle through the three given points in each of the following cases:
(i) (4, 6), (-2, 4), (8, -6)."

I proceeded following the example of the textbook:

Let (4, 6) be A, let (-2, 4) be B and let (8, -6) be C.

The midpoint of AB is (1, 5); the gradient of AB is 1/3.

The midpoint of BC is (3, -1); the gradient of BC is -1.

Therefore, the equation of the perpendicular bisector of AB is:

Therefore also, the equation of the perpendicular bisector of BC is:

Eliminate y:

Sub x into (2):

Therefore, the centre of the circle is (-2, 4).

Let the radius of the circle be r.

Therefore, the equation of the circle is:

However, the book has:

So, my question for you is: which of us is wrong?

Ta

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the equation of a circle

Hi;

I have checked the book answer, it is correct.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Finding the equation of a circle

Aha! I must have made a mistake somewhere then; though quite where, I'm not sure.

Thanks for the help bobbym

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the equation of a circle

Hi;

Therefore, the centre of the circle is (-2, 4).

The point (-2,4) is one of your given points that lie on the circle, it can not of course be on the circle and the center of the circle.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Finding the equation of a circle

You're right!

I went ahead and used the gradients of AB and BC when finding the equations of the perpendicular bisectors of these lines. Of course, I should have used

Where m is the gradient of the perpendicular bisector of the line and

is the gradient of the line.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Finding the equation of a circle

Hi;

The next time you are confronted with two different solutions remember that each point given is a solution to the equation of the circle. So all you need to do to verify correctness is to plug in.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.