Finding the Equation of a Circle

When we’re tasked with finding the equation of a circle, we’re essentially trying to fill in the blanks of the standard form: [latex](x - h)^2 + (y - k)^2 = r^2[/latex]. Here, [latex](h, k)[/latex] represents the center of the circle, and [latex]r[/latex] is its radius. But how do we proceed when we’re given different pieces of information? Let’s explore this through various scenarios.

We might encounter three scenarios:

1. We know the center and radius
2. We know the center and a point on the circle
3. We know two points on the circle (often the diameter)

So far all the problems we have worked with fall into the first scenario, but that will not always be the case. Let’s break down how to find the equation of a circle in standard form given the other two scenarios.

The Center and a Point

Let’s say you know the center of the circle and one point on its circumference. This is akin to knowing where you’re standing (the center) and spotting a friend waving from the edge of a circular field. To find the equation, you’ll use the distance formula to calculate the radius.

[latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/latex]

The distance between the center [latex](h, k)[/latex] and your point [latex](x, y)[/latex] is the radius. Once you have this, you can complete the standard form equation. It’s like measuring the distance to your friend to figure out how big the field is.

Two Points on the Circle

Now, what if we don’t know the center either? Imagine your friend has brought along another buddy, and they’re standing on opposite sides of the circular field. This scenario builds on the previous one, but with a twist – we need to find the center first.

Picture these two friends standing on opposite sides of a circular pool. To find the equation, you’ll first need to determine the center. The good news is that the line connecting your two friends is actually the diameter of the circle. The midpoint between these two points is your center.

You can find the center by using the midpoint formula, which is like finding the middle point between two ends:

[latex]\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)[/latex]

Once you’ve found the center this way, you’re back in familiar territory. You can now use the distance formula we learned in the previous scenario to calculate the radius. Simply measure the distance from your newly found center to either of your friends’ positions. With the center and radius now in hand, writing the circle’s equation becomes a piece of cake!