A circle of radius 5 passes the points 0,0 and 8,0 find the two possible centres of the circle

To find the two possible centers of a circle that passes through the points \((0, 0)\) and \((8, 0)\) with a radius of 5, let's proceed step-by-step.

### Step 1: Equation of a Circle
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius. Here, \(r = 5\).

### Step 2: Analyze the Given Points
The circle passes through the points \((0, 0)\) and \((8, 0)\). These two points lie on the x-axis, so the center \((h, k)\) must be somewhere along the perpendicular bisector of the line segment joining \((0, 0)\) and \((8, 0)\).

The midpoint of the segment joining \((0, 0)\) and \((8, 0)\) is:
\[
\left(\frac{0+8}{2}, \frac{0+0}{2}\right) = (4, 0)
\]

### Step 3: Use the Geometry of the Circle
The center must lie on the perpendicular bisector of the line joining \((0, 0)\) and \((8, 0)\). The perpendicular bisector of a horizontal line segment is a vertical line through the midpoint \((4, 0)\), meaning the center must be at some point \((4, k)\).

### Step 4: Solve for the Centers
Since the radius is 5, the distance between the center and either point (for example, \((0, 0)\)) must be 5. Using the distance formula between \((4, k)\) and \((0, 0)\), we have:
\[
\sqrt{(4-0)^2 + (k-0)^2} = 5
\]
Simplifying:
\[
\sqrt{16 + k^2} = 5
\]
Squaring both sides:
\[
16 + k^2 = 25
\]
\[
k^2 = 9
\]
\[
k = \pm 3
\]

### Step 5: The Two Possible Centers
Thus, the two possible centers of the circle are:
\[
(4, 3) \quad \text{and} \quad (4, -3)
\]

These are the two possible centers of the circle.

Would you like further details or have any questions?

---

Here are 5 related questions to explore further:
1. How do we find the equation of the circle given one of these centers?
2. What if the radius of the circle was different? How would that change the result?
3. What are the equations of the two circles that pass through the points \((0,0)\) and \((8,0)\)?
4. How would you find the center if the radius were unknown but another point on the circle was given?
5. What if the points were not on the x-axis? How would you approach the problem?

**Tip:** When dealing with circles, the perpendicular bisector of any chord passes through the center of the circle.

A circle of radius 5 passes the points 0,0 and 8,0. Find the two possible centers of the circle.

Learn how to find the two possible centers of a circle that passes through the points (0,0) and (8,0) with a radius of 5. This problem covers key geometry concepts, including the equation of a circle, distance formula, and perpendicular bisector theorem. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation