Answer:
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 .