can you do b again but use the stand formula aand the gernal formula

Let's redo part (b) and provide both the standard form and the general form of the circle's equation.

### Circle with diameter endpoints \(A(-6, 4)\) and \(B(5, -4)\)

First, find the center \((h, k)\) of the circle:
\[
h = \frac{x_1 + x_2}{2} = \frac{-6 + 5}{2} = \frac{-1}{2}
\]
\[
k = \frac{y_1 + y_2}{2} = \frac{4 - 4}{2} = 0
\]
So, the center is \(\left(-\frac{1}{2}, 0\right)\).

Next, find the radius \(r\):
\[
r = \frac{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}{2} = \frac{\sqrt{(5 - (-6))^2 + (-4 - 4)^2}}{2}
\]
\[
r = \frac{\sqrt{(11)^2 + (-8)^2}}{2} = \frac{\sqrt{121 + 64}}{2} = \frac{\sqrt{185}}{2}
\]

**Standard Form:**
\[
\left(x + \frac{1}{2}\right)^2 + y^2 = \left(\frac{\sqrt{185}}{2}\right)^2
\]
\[
\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{185}{4}
\]

**General Form:**

Expand the standard form equation:
\[
\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{185}{4}
\]
\[
\left(x^2 + x + \frac{1}{4}\right) + y^2 = \frac{185}{4}
\]
Multiply through by 4 to clear the fraction:
\[
4x^2 + 4x + 1 + 4y^2 = 185
\]
\[
4x^2 + 4y^2 + 4x + 1 = 185
\]
Subtract 185 from both sides to set the equation to 0:
\[
4x^2 + 4y^2 + 4x + 1 - 185 = 0
\]
\[
4x^2 + 4y^2 + 4x - 184 = 0
\]

### Summary of the Equation

- **Standard Form:** \(\left(x + \frac{1}{2}\right)^2 + y^2 = \frac{185}{4}\)
- **General Form:** \(4x^2 + 4y^2 + 4x - 184 = 0\)

Would you like more details or have any questions?

### Relative Questions

1. How do you determine the radius of a circle given its diameter?
2. What steps are involved in converting a circle's equation from standard to general form?
3. How do you find the center of a circle given two endpoints of its diameter?
4. Why do you multiply by 4 when converting the standard form to the general form in this problem?
5. What is the significance of the general form of a circle's equation?
6. How can the distance formula be used to find the radius of a circle?
7. What are the differences between standard form and general form of a circle's equation?
8. How do you expand binomials when converting from standard to general form?

### Tip

When working with circles, always ensure to double-check the calculations for the radius and the center, as small errors can lead to incorrect equations.

Learn how to find the equation of a circle given its diameter endpoints. This problem covers key concepts in circle geometry and coordinate geometry, including the distance formula and the equation of a circle in both standard and general forms. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation