Understanding the Equation of a Circle
The equation of a circle in standard form is given by:
(x - h)2 (y - k)2 r2
where (h, k) is the center of the circle and r is the radius.
Problem and Solution
Given a circle with center at (0, 4) and passing through the point (0, 5/2), we need to find the equation of the circle in standard form.
Step 1: Calculate the Radius
The radius can be calculated using the distance formula between the center and the given point.
The distance formula is:
Distance u221A(x2 - x1)2 (y2 - y1)2
Here, the center is (0, 4) and the point is (0, 5/2).
Distance u221A(0 - 0)2 (5/2 - 4)2
Distance u221A(0) (5/2 - 8/2)2
Distance u221A(-3/2)2
Distance u221A9/4 3/2
Step 2: Substitute the Values into the Standard Form
Now that we have the radius, we can substitute h, k, and r into the standard form equation.
The center is (0, 4) and the radius is 3/2.
(x - 0)2 (y - 4)2 (3/2)2
x2 (y - 4)2 9/4
Additional Information
Always draw a diagram to focus your thoughts and think logically. Here is the representation of the circle:
Calculating the expanded form of the circle equation:
x2 (y2 - 8y 16) 9/4
x2 y2 - 8y 16 9/4
x2 y2 - 8y (64/4 - 9/4) 0
x2 y2 - 8y 55/4 0
Conclusion
The equation of the circle in standard form is:
x2 (y - 4)2 9/4
Or, in expanded form:
x2 y2 - 8y 55/4 0