Consider the following equation of a circle.

(x−12)2+y2=2516(x−12)2+y2=2516

Step 3 of 3: 

Find three points that lie on the circle.

To find three points that lie on the circle described by the equation:

\[
(x - 12)^2 + y^2 = \frac{25}{16}
\]

we can substitute specific \(x\) or \(y\) values into the equation and solve for the corresponding \(y\) or \(x\) values. Let's proceed with this method.

### 1. **Substituting \(x = 12\):**

\[
(12 - 12)^2 + y^2 = \frac{25}{16}
\]

This simplifies to:

\[
0 + y^2 = \frac{25}{16}
\]

\[
y^2 = \frac{25}{16}
\]

Taking the square root:

\[
y = \pm \frac{5}{4}
\]

So, the points are \((12, \frac{5}{4})\) and \((12, -\frac{5}{4})\).

### 2. **Substituting \(y = 0\):**

\[
(x - 12)^2 + 0 = \frac{25}{16}
\]

This simplifies to:

\[
(x - 12)^2 = \frac{25}{16}
\]

Taking the square root:

\[
x - 12 = \pm \frac{5}{4}
\]

Solving for \(x\):

\[
x = 12 \pm \frac{5}{4}
\]

So, the points are \(\left(12 + \frac{5}{4}, 0\right)\) and \(\left(12 - \frac{5}{4}, 0\right)\).

Simplifying:

\[
\left(\frac{53}{4}, 0\right)\quad \text{and}\quad \left(\frac{43}{4}, 0\right)
\]

### Three Points on the Circle:

1. \(\left(12, \frac{5}{4}\right)\)
2. \(\left(12, -\frac{5}{4}\right)\)
3. \(\left(\frac{53}{4}, 0\right)\)

Would you like to explore any of these points further or have any additional questions?

Here are some related questions:

1. How do you determine whether a point lies on a given circle?
2. What are the properties of a circle in the Cartesian plane?
3. How can you graph a circle using its equation?
4. What is the significance of the radius in the equation of a circle?
5. How do transformations affect the equation of a circle?

**Tip:** When selecting points on a circle, it's often easiest to choose points aligned with the axes (i.e., where either \(x\) or \(y\) is zero) to simplify calculations.

Learn how to find points on the circle described by the equation (x - 12)^2 + y^2 = 25/16. Step-by-step solution includes substituting specific x or y values to determine points on the circle. Suitable for Grades 10-12 students studying coordinate geometry and circles.

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