https://file.aiquickdraw.com/mathbot/file/1725375993515i4vgr83j.jpg

The image contains the following problem:

"Find the conjugate point of \( (2, 3) \) with respect to the circle \( x^2 + y^2 = 25 \)."

### Solution

To find the conjugate point of a given point \( (x_1, y_1) \) with respect to a circle of the form \( x^2 + y^2 = r^2 \), we use the formula:

\[
(x', y') = \left( \frac{r^2 x_1}{x_1^2 + y_1^2}, \frac{r^2 y_1}{x_1^2 + y_1^2} \right)
\]

Given:
- Point: \( (x_1, y_1) = (2, 3) \)
- Circle: \( x^2 + y^2 = 25 \), so \( r = 5 \)

Now, let's substitute the values:

1. Compute \( x_1^2 + y_1^2 = 2^2 + 3^2 = 4 + 9 = 13 \).
2. Use the formula:

\[
(x', y') = \left( \frac{5^2 \cdot 2}{13}, \frac{5^2 \cdot 3}{13} \right)
\]

\[
(x', y') = \left( \frac{25 \cdot 2}{13}, \frac{25 \cdot 3}{13} \right)
\]

\[
(x', y') = \left( \frac{50}{13}, \frac{75}{13} \right)
\]

### Conclusion
The conjugate point of \( (2, 3) \) with respect to the circle \( x^2 + y^2 = 25 \) is \( \left( \frac{50}{13}, \frac{75}{13} \right) \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the conjugate point of a point with respect to an ellipse?
2. What is the geometric interpretation of the conjugate point concerning a circle?
3. How can this concept be extended to three-dimensional geometry?
4. How do you determine if a point is inside or outside a circle using its equation?
5. What are the applications of conjugate points in physics or engineering?

**Tip:** Remember that the conjugate point concept can also be applied to other conic sections like ellipses and hyperbolas, with similar but slightly different formulas!

Find the conjugate point of (2, 3) with respect to the circle x^2 + y^2 = 25.

Learn how to find the conjugate point of (2, 3) with respect to the circle x^2 + y^2 = 25. This problem involves key concepts in analytic geometry, including the conjugate point theorem and the application of circle equations. 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