https://file.aiquickdraw.com/mathbot/file/1725494572104jkuo24ny.jpg

The problem asks for the equation of the tangent line to a circle that passes through a specific point, labeled \( C \), on the graph.

### Analyzing the Image:

1. **Circle**: The circle is centered at the origin \((0,0)\) and seems to have a radius of 5 units. This can be inferred from the circle's intersection with the y-axis at \((0, 5)\) and \((0, -5)\).
  
   Therefore, the equation of the circle is:
   \[
   x^2 + y^2 = 25
   \]

2. **Point \( C \)**: The point \( C \) is located at \((3, 4)\), which is on the circle since \( 3^2 + 4^2 = 9 + 16 = 25 \), satisfying the circle's equation.

### Finding the Tangent Line Equation:

For a circle centered at the origin, the equation of the tangent line to the circle at a point \((x_1, y_1)\) can be given by:
\[
x_1x + y_1y = r^2
\]
Substituting the coordinates of \( C(3, 4) \) and the radius \( r = 5 \) into this equation:
\[
3x + 4y = 25
\]

### Conclusion:

The equation of the tangent line to the circle at point \( C(3, 4) \) is:
\[
3x + 4y = 25
\]

Would you like more details or have any questions?

Here are 5 related questions that expand on this information:
1. How do you derive the equation of a tangent line for a circle with a different center?
2. What happens if point \( C \) is not on the circle? How would you find the equation of the tangent line then?
3. Can you explain the geometric significance of the slope of the tangent line?
4. What are the equations of the tangent lines that pass through a point outside the circle?
5. How would you find the point of tangency if given only the equation of the tangent line and the circle?

**Tip:** The slope of the tangent to a circle at any point is perpendicular to the radius at that point. This can be a useful property when solving more complex geometry problems.

What is the equation of the tangent line to the circle passing through point C in the image?

Find the equation of the tangent line to the circle centered at the origin with a radius of 5, passing through the point C(3, 4). This problem involves key concepts of geometry and analytic geometry. Suitable for students in Grades 10-12.

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