P(2,3) is on the circle with center O(0,0) 
a) write an equation of the circle
b) write the equation of a tangent to the circle

Let's solve each part step-by-step:

### a) Equation of the Circle

The general equation of a circle with center \( O(h, k) \) and radius \( r \) is:

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

For this problem, the center of the circle is \( O(0, 0) \). Thus, the equation simplifies to:

\[
x^2 + y^2 = r^2
\]

To find the radius \( r \), we use the distance formula between the center \( O(0,0) \) and the point \( P(2,3) \):

\[
r = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

Therefore, the equation of the circle is:

\[
x^2 + y^2 = 13
\]

### b) Equation of a Tangent to the Circle

The equation of a tangent to a circle at a point \( (x_1, y_1) \) on the circle with center at \( (0, 0) \) is given by:

\[
x_1x + y_1y = r^2
\]

Since \( P(2, 3) \) is a point on the circle, \( x_1 = 2 \) and \( y_1 = 3 \). The radius \( r^2 = 13 \), so the equation of the tangent at point \( P(2, 3) \) is:

\[
2x + 3y = 13
\]

### Summary

- **Equation of the Circle:** \( x^2 + y^2 = 13 \)
- **Equation of the Tangent at \( P(2, 3) \):** \( 2x + 3y = 13 \)

Would you like more details or have any questions?

### Related Questions:

1. How to derive the formula for the tangent line to a circle?
2. What is the distance formula, and how is it derived?
3. How to find the equation of a circle given its diameter's endpoints?
4. What are the properties of a tangent to a circle?
5. How to find the intersection points of a line and a circle?

### Tip:

To quickly check if a point lies on a circle, substitute the point's coordinates into the circle's equation and see if it satisfies it.

P(2,3) is on the circle with center O(0,0). a) Write an equation of the circle. b) Write the equation of a tangent to the circle.

Learn how to find the equation of a circle with center O(0,0) and a point P(2,3) on the circle. The step-by-step solution includes using the distance formula to determine the radius and finding the equation of a tangent line at point P(2,3). Suitable for Grades 9-11.

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