1 The coovdinate of the end point of a diancia of 1 circle zre and Find
2. Equation of the curcle.

Lengtis of the diameter
c Equation of the diameter

Ask by Lindsey Lawson. in Nigeria

Jan 22,2025

Question

1 The coovdinate of the end point of a diancia of 1 circle zre and Find
2. Equation of the curcle.

Lengtis of the diameter
c Equation of the diameter

Ask by Lindsey Lawson. in Nigeria

Jan 22,2025

UpStudy AI · Accepted Answer

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!
**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:** $$ (x + 1)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{113}{4} $$

1 The coovdinate of the end point of a diancia of 1 circle zre and Find
2. Equation of the curcle.

Lengtis of the diameter
c Equation of the diameter

Upstudy AI Solution

Answer

Solution

Problem Statement

1. Length of the Diameter

Using the Distance Formula:

Plugging in the Values:

Result:

2. Equation of the Diameter

Steps to Find the Equation:

a. Find the Slope ()

b. Use Point-Slope Form

c. Convert to Slope-Intercept Form ()

Alternative: Standard Form

Result:

3. Equation of the Circle

a. Finding the Center ()

b. Finding the Radius ()

c. Writing the Equation of the Circle

Final Equation:

Summary

Mind Expander

Related Questions

Latest Geometry Questions

1 The coovdinate of the end point of a diancia of 1 circle zre and Find 2. Equation of the curcle. Lengtis of the diameter c Equation of the diameter

Upstudy AI Solution

Answer

Solution

Problem Statement

1. Length of the Diameter

Using the Distance Formula:

Plugging in the Values:

Result:

2. Equation of the Diameter

Steps to Find the Equation:

a. Find the Slope ( )

b. Use Point-Slope Form

c. Convert to Slope-Intercept Form ( )

Alternative: Standard Form

Result:

3. Equation of the Circle

a. Finding the Center ( )

b. Finding the Radius ( )

c. Writing the Equation of the Circle

Final Equation:

Summary

Mind Expander

Related Questions

Latest Geometry Questions

1 The coovdinate of the end point of a diancia of 1 circle zre and Find
2. Equation of the curcle.

Lengtis of the diameter
c Equation of the diameter

a. Find the Slope ()

c. Convert to Slope-Intercept Form ()

a. Finding the Center ()

b. Finding the Radius ()