Equation of a Circle with Radius 5 cm and Center on the y-axis
Given a circle with a radius of 5 cm and a center on the y-axis, we can determine its equation. Let's go through the steps to find the exact equation, including an example where the circle passes through the point (3, 2).
Identifying the Center
Since the center lies on the y-axis, we can denote the center as (0, k), where k is the y-coordinate we need to determine.
Using the Radius
The equation of a circle with center (h, k) and radius r is given by:
[ (x - h)^2 (y - k)^2 r^2 ]For our circle, the equation becomes:
[ x^2 (y - k)^2 25 ]Substituting the Given Point
We know the circle passes through the point (3, 2). Thus, we can substitute x 3 and y 2 into the equation:
[ 3^2 (2 - k)^2 25 ]This simplifies to:
[ 9 (2 - k)^2 25 ]Further simplification gives:
[ (2 - k)^2 16 ]We need to solve for k. Taking the square root of both sides, we get two possible equations:
[ 2 - k 4 quad text{or} quad 2 - k -4 ]Solving these equations:
- From 2 - k 4, we get k -2 - From 2 - k -4, we get k 6We now have two possible centers: (0, -2) and (0, 6).
Equations of the Circles
For the center (0, -2) and (0, 6), the corresponding equations of the circles are:
[ x^2 (y 2)^2 25 ] [ x^2 (y - 6)^2 25 ]These equations represent two circles that satisfy the given conditions: a radius of 5 cm and a center on the y-axis, with an additional condition that they pass through the point (3, 2).
Conclusion
We have determined the equations of the two circles based on the given conditions. This problem is useful for understanding how to apply the circle equation in specific scenarios, such as when the center lies on the y-axis.