https://file.aiquickdraw.com/mathbot/file/1723171426064062z2pja.jpg

Write the standard equation of a circle whose center

The image shows a handwritten problem asking to "Write the standard equation of the circle whose center is at \((-3, 5)\) and passes through the point \((1, 2)\)."

### Solution:

1. **Identify the center and a point on the circle**:
   - Center \((h, k) = (-3, 5)\)
   - Point on the circle \((x_1, y_1) = (1, 2)\)

2. **Formula for the standard equation of a circle**:
   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]
   where \( r \) is the radius.

3. **Find the radius**:
   The radius \( r \) is the distance between the center \((h, k)\) and the given point \((x_1, y_1)\), calculated by:
   \[
   r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
   \]
   Substituting the given values:
   \[
   r = \sqrt{(1 - (-3))^2 + (2 - 5)^2} = \sqrt{(1 + 3)^2 + (-3)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
   \]

4. **Write the equation**:
   Substitute \( h = -3 \), \( k = 5 \), and \( r = 5 \) into the standard equation of the circle:
   \[
   (x - (-3))^2 + (y - 5)^2 = 5^2
   \]
   Simplifying this:
   \[
   (x + 3)^2 + (y - 5)^2 = 25
   \]

### Final Answer:
The standard equation of the circle is:
\[
(x + 3)^2 + (y - 5)^2 = 25
\]

Would you like more details or have any questions?

---

### Related Questions:

1. What is the general form of a circle's equation?
2. How do you find the radius if only the diameter is given?
3. What are the steps to convert the general form to the standard form of a circle's equation?
4. How can you find the center and radius from the general form of a circle's equation?
5. What is the equation of a circle if its diameter endpoints are given?
6. How does changing the center coordinates affect the circle's equation?
7. How do you find the equation of a circle if it is tangent to the x-axis?
8. How do you determine if a point lies inside, on, or outside a given circle?

### Tip:
When given the center and a point on a circle, always start by finding the radius using the distance formula before substituting into the standard circle equation.

Learn how to write the standard equation of a circle given its center at (-3, 5) and passing through the point (1, 2). This problem involves using the distance formula and the standard equation of a circle. Suitable for high school students studying geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation