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?

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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.