Can You Find the Equation of a Circle That Touches the Coordinate Axes and Whose Center Lies on a Given Line?
When working with circles in the coordinate plane, it is often necessary to find the equation of a circle that not only touches the coordinate axes but also has its center lying on a specific line. This problem combines concepts from coordinate geometry and algebra to determine the correct circle's equation. In this article, we'll explore the step-by-step process to find such a circle.
Understanding the Properties of the Circle
A circle that touches both the x-axis and y-axis has its center at a point (r, r), where r is the radius of the circle. This is because the distance from the center to each axis must be equal to the radius. The equation of a circle with radius r and center at (h, k) is given by:
[ (x - h)^2 (y - k)^2 r^2 ]In this specific problem, we are given the line equation:
[ x - 2y 3 ]We need to find the equation of the circle whose center lies on this line and that touches the coordinate axes.
Expressing the Center
Let's denote the center of the circle as (r, r) since it touches both axes. Substituting these coordinates into the line equation:
[ r - 2r 3 ]Simplifying this:
[ -r 3 implies r -3 ]However, since r represents the radius, it must be a positive value. This result does not make sense in the context of a circle. Let's re-examine the problem from another perspective.
Revisiting the Center and Using the Line Equation
The center of the circle (x0, y0) must satisfy the line equation:
[ x_0 - 2y_0 3 ]Expressing the center in terms of a variable point on the line:
[ x_0 2y_0 3 ]Since the circle touches the coordinate axes, the center must be equidistant from both axes. Thus:
[ x_0 y_0 ]Substituting y_0 x_0 into the line equation:
[ x_0 - 2x_0 3 implies -x_0 3 implies x_0 -3 ]This implies that both x_0 and y_0 are -3. Therefore, the center of the circle is (-3, -3).
Finding the Radius
The radius r is the distance from the center to either axis:
[ r -3 3 3 ]Equation of the Circle
Using the standard form of the equation of a circle:
[ (x - h)^2 (y - k)^2 r^2 ]Where h -3, k -3, and r 3:
[ (x 3)^2 (y 3)^2 9 ]Thus, the equation of the circle is:
[box] (x 3)^2 (y 3)^2 9 [/box]By following these steps, we have successfully determined the equation of the circle that touches the coordinate axes and whose center lies on the line x - 2y 3.
Conclusion
Understanding the properties of circles in the coordinate plane and combining them with the given line equation is crucial in solving problems like this. The key steps include identifying the center of the circle, using the line equation, and ensuring the circle's radius satisfies both the given conditions.
Additional Information
You can apply similar methods to find the equation of a circle touching the axes in other contexts or with different line equations. For further practice, consider exploring more problems involving circles and coordinate geometry.