Question One

i. Find the equation of circle with the coordinates (-2, 3) and radius 6.

To find the equation of a circle given its center and radius, we use the standard form of the circle's equation:

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

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

Given:
- Center \((h, k) = (-2, 3)\)
- Radius \(r = 6\)

Substitute these values into the standard form equation:

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

Simplify the equation:

\[ (x + 2)^2 + (y - 3)^2 = 36 \]

So, the equation of the circle is:

\[ (x + 2)^2 + (y - 3)^2 = 36 \]

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert the general form of a circle's equation to its standard form?
2. What is the distance formula and how is it related to finding the radius of a circle?
3. How do you find the center and radius of a circle given its equation in general form?
4. What are the properties of a circle in the coordinate plane?
5. How do you determine if a point lies inside, outside, or on a circle?
6. How can you graph a circle given its equation?
7. What transformations can be applied to the equation of a circle?
8. How do you find the intersection points of two circles?

**Tip:** To quickly identify the center and radius from the equation \((x - h)^2 + (y - k)^2 = r^2\), note that the values of \(h\) and \(k\) are the coordinates of the center with opposite signs from those in the equation, and \(r\) is the square root of the number on the right side.

Learn how to find the equation of a circle given its center and radius. Step-by-step solution to determine the equation using the standard form of a circle's equation. Suitable for high school students studying coordinate geometry.

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