https://file.aiquickdraw.com/mathbot/file/17273121989983wt0doyy.jpg

The problem is asking for the equation of a circle with center \((-3, 2)\) and radius \(4\). The general equation of a circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where:
- \((h, k)\) is the center of the circle, and
- \(r\) is the radius.

### Step 1: Substituting the center and radius into the equation
Here, the center is \((-3, 2)\) and the radius is \(4\). So, substituting these values into the equation, we get:

\[
(x + 3)^2 + (y - 2)^2 = 16
\]

### Step 2: Expanding the equation
Now, let's expand both terms:

\[
(x + 3)^2 = x^2 + 6x + 9
\]
\[
(y - 2)^2 = y^2 - 4y + 4
\]

So, the equation becomes:

\[
x^2 + 6x + 9 + y^2 - 4y + 4 = 16
\]

### Step 3: Simplifying the equation
Combine the constants on the left-hand side:

\[
x^2 + 6x + y^2 - 4y + 13 = 16
\]

Subtract \(16\) from both sides:

\[
x^2 + 6x + y^2 - 4y - 3 = 0
\]

### Step 4: Compare with the options
This equation matches option **a**:
\[
x^2 + y^2 + 6x - 4y - 3 = 0
\]

### Final Answer: The correct option is **a**.

Would you like further details or clarification? Here are some related questions you can explore:

1. What is the general form of a circle equation?
2. How do you find the equation of a circle given its center and a point on it?
3. How do you complete the square to rewrite a circle equation in standard form?
4. What is the difference between the standard and general forms of a circle's equation?
5. How can you determine the center and radius from a general circle equation?

**Tip:** Always verify your expanded terms carefully when dealing with circle equations!

Pesamaan lingkaran dengan pusat (-3,2) dan jari-jari 4 adalah...

Learn how to find the equation of a circle with center (-3, 2) and radius 4. This problem covers key geometry and algebra concepts, including the circle equation and expanding quadratic terms. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation