Converting a circle equation from its standard form to general form is pretty easy once you get the hang of it. Let's go through it step by step!

Standard Form of a Circle

First, let’s remember what the standard form of a circle’s equation looks like. It is written as:

$(x - h)^2 + (y - k)^2 = r^2$

In this equation:

• $(h, k)$ is the center of the circle.
• $r$ is the radius (how far the circle stretches from the center).

General Form of a Circle

The general form of a circle’s equation looks like this:

$Ax^2 + Ay^2 + Dx + Ey + F = 0$

Now, we need to change the standard form into this general form.

Step 1: Expand the Equation

Start with the standard form:

$(x - h)^2 + (y - k)^2 = r^2$

Now we will expand (or open up) both sides of the equation:

$(x^2 - 2hx + h^2) + (y^2 - 2ky + k^2) = r^2$

Then, combine everything together:

$x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2$

Step 2: Rearrange to General Form

Next, we move $r^2$ to the left side of the equation:

$x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0$

Now, if you look closely, it can be arranged like this:

$x^2 + y^2 - 2hx - 2ky + F = 0$

Where $F$ is $h^2 + k^2 - r^2$.

Step 3: Identify Coefficients

Now, let’s find out what the coefficients are in this general form:

• $A = 1$ (this is the number in front of $x^2$ and $y^2$).
• $D = -2h$ (this comes from the $x$ term).
• $E = -2k$ (this comes from the $y$ term).
• $F = h^2 + k^2 - r^2$.

Conclusion

That’s it! You’ve just changed the circle's equation from standard form to general form. It might feel a bit tricky at first, but don’t worry! With practice, it will become easier for you. Just take it one step at a time. Happy studying!