Write the general equation for the circle that passes through the points:
(-1, 2)
(4, 2)
(-3, 4)

General form:

Standard form:

Step-by-step explanation:

The standard form for a circle is where is the center and is the radius.

Our goal is to write our equation in general form: .

So I'm going to use standard form right now to try and see if it helps.

Plug in the point :

           Equation 1

Plug in the point :

           Equation 2

Plug int he point :

          Equation 3

I notice Equation 1 and Equation 2 will have a lot of stuff to cancel if I chose to do Equation 1 minus Equation 2. So let's do that.

Add on both sides:

The implies:

This us gives us two equation to solve for :

or

The first equation says -1=4 which is never true so we will solve the second one.

Add on both sides:

Add 4 on both sides:

Divide 2 on both sides:

So let's look at Equation 2 and Equation 3 with applied to them:

Let's simplify them a bit by performing the addition/subtraction in the ( ):

Now a little more by applying the square:

I will subtract these two equations now because I see it will give an equation just in terms of to solve:

Expand the binomial squares using the identity :

Distribute:

Combine like terms:

Simplify:

Add 26 on both sides:

Divide both sides by 4:

Reduce:

So we now have the center of the circle . We have multiple points to choose from so that we can find the radius. (We will find the radius by finding the distance from the center of the circle to a point on the circle.)

Let's find the the distance from and .

You may use Distance Formula or Pythagorean Theorem.

Find the horizontal distance of the triangle: .

Find the vertical distance of the triangle: .

Now we will find the hypotenuse,, using .

Simplify the squares:

Simplify by adding:

(This is ; we don't need to find , but I will.)

-Unnecessary for the problem; finding the radius, -

Take the square root of both sides:

Simplify the square root:

The equation in standard form is:

(or if you simplify the fraction on the right: .)

Now we wanted this in general form so we will need to expand the binomial squares:

Multiply both sides by 4 to get rid of the fractions:

Reorder to put in order using commutative property:

Simplify the addition on 169 and 9:

Subtract 106 on both sides:

The general form is .