Equation of a Circle Touching Parallel Lines
In this article, we explore the geometric relationship between a circle and parallel lines. Specifically, we derive the equation of a circle that touches the y-axis and the line xc, thereby touching two parallel lines.
Understanding the Problem
We are given two parallel lines defined by the equations:
x 0 - the y-axis x cSince the circle touches both these lines, it must be positioned between them. This implies that the circle will have a radius that is half the distance between the two lines. The center of the circle will lie on the line that is halfway between the two given lines.
Geometric Analysis
1. Calculation of Radius: - The distance between the lines x 0 and x c is simply c units. - Therefore, the radius of the circle, r, is half of this distance: r frac{c}{2}.
2. Position of the Center: - The center of the circle must lie on the line halfway between the y-axis and the line x c. Hence, the x-coordinate of the center is frac{c}{2}. Let the y-coordinate of the center be y.
3. Distance from the X-axis: - Since the circle touches the X-axis, the y-coordinate of the center must be equal to the radius of the circle, i.e., y pm frac{c}{2}.
Equation of the Circle
The general equation of a circle with center at (h, k) and radius r is:
(x - h)^2 (y - k)^2 r^2
Substituting the values for our specific circle:
1. Circle 1 (Center above X-axis):
(x - frac{c}{2})^2 (y - frac{c}{2})^2 (frac{c}{2})^2
2. Circle 2 (Center below X-axis):
(x - frac{c}{2})^2 (y frac{c}{2})^2 (frac{c}{2})^2
Conclusion
In summary, we have derived the equations of two circles that touch the y-axis and the line x c. The equations are based on the geometric properties of circles touching lines, which are fundamental concepts in analytic geometry.