Answer
**Length of the Diameter:** \( \sqrt{113} \) units (approximately 10.63 units)
**Equation of the Diameter:**
- Slope-Intercept Form: \( y = -\frac{7}{8}x - \frac{11}{8} \)
- Standard Form: \( 7x + 8y + 11 = 0 \)
**Equation of the Circle:**
\[
(x + 1)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{113}{4}
\]
Solution
Certainly! Let's break down the problem step by step.
---
## **Problem Statement**
Given the endpoints of a diameter of a circle:
- Point \( A(-5, 3) \)
- Point \( B(3, -4) \)
**Tasks:**
1. Find the length of the diameter.
2. Find the equation of the diameter.
3. Determine the equation of the circle.
---
## **1. Length of the Diameter**
The length of the diameter is the distance between points \( A(-5, 3) \) and \( B(3, -4) \).
### **Using the Distance Formula:**
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### **Plugging in the Values:**
\[
\text{Diameter Length} = \sqrt{(3 - (-5))^2 + (-4 - 3)^2} = \sqrt{(8)^2 + (-7)^2} = \sqrt{64 + 49} = \sqrt{113}
\]
### **Result:**
\[
\text{Diameter Length} = \sqrt{113} \quad \text{units} \approx 10.63 \quad \text{units}
\]
---
## **2. Equation of the Diameter**
The diameter is the straight line connecting points \( A(-5, 3) \) and \( B(3, -4) \).
### **Steps to Find the Equation:**
### **a. Find the Slope (\( m \))**
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 3}{3 - (-5)} = \frac{-7}{8}
\]
### **b. Use Point-Slope Form**
Using point \( A(-5, 3) \):
\[
y - y_1 = m(x - x_1)
\]
\[
y - 3 = -\frac{7}{8}(x + 5)
\]
### **c. Convert to Slope-Intercept Form (\( y = mx + c \))**
\[
y = -\frac{7}{8}x - \frac{35}{8} + 3
\]
\[
y = -\frac{7}{8}x - \frac{35}{8} + \frac{24}{8}
\]
\[
y = -\frac{7}{8}x - \frac{11}{8}
\]
### **Alternative: Standard Form**
Multiply both sides by 8 to eliminate fractions:
\[
8y = -7x - 11
\]
\[
7x + 8y + 11 = 0
\]
### **Result:**
- **Slope-Intercept Form:** \( y = -\frac{7}{8}x - \frac{11}{8} \)
- **Standard Form:** \( 7x + 8y + 11 = 0 \)
---
## **3. Equation of the Circle**
To find the equation of the circle, we need:
- **Center (\( C(h, k) \))**
- **Radius (\( r \))**
### **a. Finding the Center (\( C \))**
The center is the midpoint of the diameter.
\[
C\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
\[
C\left( \frac{-5 + 3}{2}, \frac{3 + (-4)}{2} \right) = C\left( \frac{-2}{2}, \frac{-1}{2} \right) = C(-1, -0.5)
\]
### **b. Finding the Radius (\( r \))**
The radius is half the length of the diameter.
\[
r = \frac{\sqrt{113}}{2}
\]
### **c. Writing the Equation of the Circle**
**Standard Form of a Circle:**
\[
(x - h)^2 + (y - k)^2 = r^2
\]
**Plugging in the Values:**
\[
(x - (-1))^2 + \left( y - (-0.5) \right)^2 = \left( \frac{\sqrt{113}}{2} \right)^2
\]
\[
(x + 1)^2 + (y + 0.5)^2 = \frac{113}{4}
\]
### **Final Equation:**
\[
(x + 1)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{113}{4}
\]
---
## **Summary**
1. **Length of the Diameter:**
\[
\sqrt{113} \quad \text{units} \approx 10.63 \quad \text{units}
\]
2. **Equation of the Diameter:**
- **Slope-Intercept Form:** \( y = -\frac{7}{8}x - \frac{11}{8} \)
- **Standard Form:** \( 7x + 8y + 11 = 0 \)
3. **Equation of the Circle:**
\[
(x + 1)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{113}{4}
\]
---
If you have any further questions or need additional clarification on any of the steps, feel free to ask!
Reviewed and approved by the UpStudy tutoring team
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