Circles - Find Standard Equation Given Two Points

Circles - Find Standard Equation Given Two Points - How it Works - Video

Example 1

Example 1:

Here we are given the center (-1, 2) and the radius 2.

Our standard equation for a circle is (x - h)² + (y - k)² = r². The center point is represented as (h, k).

So h = -1 and k = 2. Now we can substitute those values into the standard equation.

Now we have (x - [-1])² + (y - 2)² = 2². Let's simplify. So our final answer is (x +1)² + (y - 2)² = 4.

Example 2

Example 2:

Here we are given the center (1.5, 0.5) and the circle is tangent to the y-axis.

Our standard equation for a circle is (x - h)² + (y - k)² = r². The center point is represented as (h, k).

So h = 1.5 and k = 0.5. Now we can substitute those values into the standard equation.

Now we have (x - 1.5)² + (y - 0.5)² = r². We don't know the radius just quite yet. We have to use the tangent to the y-axis to find that.

Since our circle is tangent to the y-axis, we know that one point on the circumference is on the y-axis. In the picture we have drawn the radius of our circle to the y-axis. As our center is at (1.5, 0.5), we can use the x point in our ordered pair to find the distance from the center to the point on the circumference, which is 1.5 or 3/2. We are going to use 3/2 because it easier to square than a decimal number.

Now let's substitute our radius into the formula => (x - 1.5)² + (y - 0.5)² = (3/2)². Let's simplify. So our final answer is (x - 1.5)² + (y - 0.5)² = 9/4.

Example 3

Example 3:

Here we are given the center (1, -1) and passes through the point (4, -1).

Our standard equation for a circle is (x - h)² + (y - k)² = r². The center point is represented as (h, k).

So h = 1 and k = -1. Now we can substitute those values into the standard equation.

Now we have (x - 1)² + (y - [-1])² = r² => (x - 1)² + (y + 1)² = r². We don't know the radius just quite yet. We have to use the point that passes through to find the radius.

We have three ways to find the radius. You can choose which one is more comfortable for you.

The first method is to substitute the point into the formula. The second method is to use the points. The third method is to use the graph and count.

The first method is below.

The second method is look at the points, (1, -1) and (4, -1). If we take a look at them, we notice that both of the y-values are the same. This means that are points form a horizontal line so we can subtract the x-values to find the distance, 4 - 1 = 3. So r = 3.

The third method is in the picture above. We have graph the points and we can count the spaces in between them, which is also 3. So r = 3.

Depending on the question one method will be faster than the others but not always.

Now, that we have found our radius, we can substitute into the formula that we have, (x - 1)² + (y + 1)² = r² => (x - 1)² + (y + 1)² = 3² . Let's simplify. So our final answer is (x - 1)² + (y + 1)² = 9.

Live Worksheet

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