CIRCLES

CONVERTING FROM GENERAL FORM TO STANDARD FORM

OBJECTIVE

• Review over standard form of a circle
• Quick review of completing the square (without solving)
• Go over general form of a circle
• Go over how to convert from general form to standard form
• Explain why it’s useful
• Do some examples
• Homework

QUICK REVIEW OVER STANDARD FORM OF A CIRCLE

Alright so to recap, we found the hypotenuse of the triangle, and named it r right?

Then, we actually found out how to find the length of r with the Pythagorean theorem.

But, what if instead of leaving the hypotenuse as part of the triangle, we detach it, and spin it around?

Well, that would make something like this:

So, what does this mean?

Well, it means that if we have a circle with a center at (0,0), and a radius r, any point (x, y) is on that circle if and only if

x² + y² = r².

Or, like we found before:

OKAY, BUT THAT SEEMS PRETTY SPECIFIC

Well, sorta, but not really.

Actually, if we really look at it, it helps us see how to find the radius of any circle (and therefore the equation).

But to do this, we need to be a little creative.

Deriving the standard form of a circle

So, we know we can use the distance formula to find the radius of a circle right?

But so far we’ve been looking at triangles that have one of their corners at (0,0).

But what if it doesn’t? Maybe something like this:

So, what we want to find is the length of d right?

And we can also see that the measure of AB = d, right?

So, then we know if we add:

(You know, because of the Pythagorean Theorem?)

But wait, if we look at the triangle, we can actually see that:

Becomes:

So, substituting, we can see that:

Which is the same as saying:

And solving for d, we get:

And there is the standard form of a circle

Building on what we just created, we can now say:

For a circle with center (h, k) and radius r, any point (x, y) is on that circle if and only if �

Squaring both sides of this equation will give you the standard form of an equation of a circle with center (h, k) and radius r :

(x – h)² + (y – k)² = r ^2.

Completing the Square Review

So, here’s how we do it.

WOW THAT’S UGLY

Yeah that looked pretty intimidating, but it’s okay, we’re going to do this in baby steps.

So for today, all we are going to look at is how to set up the equation to be solved.

We’ll worry about solving some other day.

So, let’s start off by starting some problem, say:

Example 1:

So, what we were given is:

+7 +7

Now we take 6, divide it by 2, then square it. So:

Then we add it to both sides:

Example 2:

So, what we were given is:

+64 +64

Now we take 12, divide it by 2, then square it. So:

Then we add it to both sides:

So how do we find the number that goes in the equation?

Just in case you didn’t notice (which is totally understandable).

The number that goes in:

Is:

Now this seems pretty complicated, but let’s look at another example

Example 3:

So, let’s say we are given is:

+ 75 +75

Now we take -10, divide it by 2, then square it. So:

Then we add it to both sides:

As you can see, we put a (-5) in the parenthesis, but that’s because that was:

Let’s try a few more to reinforce this.

EXAMPLE 4:

So, let’s say:

- 17 -17

Now we take 18, divide it by 2, then square it. So:

Then we add it to both sides:

As you can see, we put a (9) in the parenthesis, but that’s because that was:

General Form of a Circle

So, we’ve already touched on the general form of a circle, but to recap, the general form of a circle looks like:

​

Ax ² + By ² + Cx + Dy + E = 0

So when do we use general form?

Well, honestly, most of the time we don’t, we actually try to convert it to standard form.

However there are many times when you’re working on a problem, and a general form of a circle may pop out.

In which case, you’re gonna wanna know that what you are working with is a circle.

Trust me, it gets annoying if you don’t see it.

So, let’s get started on converting!

Converting from General to Standard

So, let’s say you are given an equation that looks like:

x ² + y ² + 6x -10y + 29 = 0

The first step to solving is the same way we complete the square.

We want to subtract 29 from both sides

-29 -29

And now we are left with:

x ² + y ² + 6x -10y = -29

Now, let’s rearrange the equation so all of our x’s and y’s are together

x ² + 6x + y ² -10y = -29

Next, we complete the square for the x side:

x ² + 6x

Remember, we take our b, divide it by 2a, and square it, then add it.

x ² + 6x + 9 + y ² -10y = -29 + 9

Now let’s complete the square for the y side:

y ² -10y

Remember again, we take our b, divide it by 2a, and square it, then add it.

(x ² + 6x + 9) + (y ² -10y + 25) = -29 + 9 + 25

Now let’s simplify

And finally we are left with:

Ugh

Yeah I know, it’s a beast, but it is how we convert from general to standard.

So to do so, basically you complete the square for both the x and y variables, and simplify that way.

So, let’s do a few more just to make sure you understand how to do it.

(And yes, you will have homework/be quizzed/tested on this).

Example 1

So, let’s say you are given an equation that looks like:

x ² + y ² + 12x -40y + 100 = 0

The first step to solving is the same way we complete the square.

We want to subtract 100 from both sides

-100 -100

And now we are left with:

x ² + y ² + 12x -40y = -100

Now, let’s rearrange the equation so all of our x’s and y’s are together

x ² + 12x + y ² -40y = -100

Next, we complete the square for the x side:

x ² + 12x

Remember, we take our b, divide it by 2a, and square it, then add it.

x ² + 12x + 36 + y ² -40y = -100 + 36

Now let’s complete the square for the y side:

y ² -40y

Remember again, we take our b, divide it by 2a, and square it, then add it.

(x ² + 12x + 36) + (y ² -40y + 400) = -100 + 36 + 400

Now let’s simplify

And finally we are left with:

Example 2

So, let’s say you are given an equation that looks like:

x ² + y ² - 24x +20y - 75 = 0

The first step to solving is the same way we complete the square.

We want to add 75 to both sides

+75 +75

And now we are left with:

x ² + y ² - 24x +20y = 75

Now, let’s rearrange the equation so all of our x’s and y’s are together

x ² - 24x + y ² +20y = 75

Next, we complete the square for the x side:

x ² - 24x

Remember, we take our b, divide it by 2a, and square it, then add it.

x ² - 24x + 144 + y ² +20y = 75 + 144

Now let’s complete the square for the y side:

y ² +20y

Remember again, we take our b, divide it by 2a, and square it, then add it.

(x ² - 24x + 144) + (y ² +20y + 100) = 75 + 144 + 100

Now let’s simplify

And finally we are left with:

Example 3

So, let’s say you are given an equation that looks like:

x ² + y ² + 14x -18y - 10 = 0

The first step to solving is the same way we complete the square.

We want to subtract 100 from both sides

+10 +10

And now we are left with:

x ² + y ² + 14x -18y = 10

Now, let’s rearrange the equation so all of our x’s and y’s are together

x ² + 14x + y ² -18y = 10

Next, we complete the square for the x side:

x ² + 14x

Remember, we take our b, divide it by 2a, and square it, then add it.

x ² + 14x + 49 + y ² -18y = 10 + 49

Now let’s complete the square for the y side:

y ² -18y

Remember again, we take our b, divide it by 2a, and square it, then add it.

(x ² + 14x + 49) + (y ² -18y + 81) = 10 + 49 + 81

Now let’s simplify

And finally we are left with: