Circle Passes Through Given Points with Center on X-Axis and Specific Radius
Understanding the equation of a circle when its center lies on the x-axis and it passes through a specific point is a fundamental concept in geometry. This article explores how to derive the equation of such a circle given a point through which it passes, a center on the x-axis, and a specified radius.
Deriving the Equation of the Circle
Consider a circle with its center at point C(h, 0) on the x-axis and a radius of 5 units. This circle passes through a point P(2, 3). To find the equation of the circle, we start by calculating the distance from the center to the point, which equals the radius.
Step 1: Calculate the Distance CP
The distance CP is given by the formula:
CP2CP2h2-(2-h)2 (3-0)252
Step 2: Solve for h
Simplifying the above expression:
h2-(2-h)2 925
h2-42-4h h216
2h2-4h-70
Solving the quadratic equation:
h4±42-4?2?(-7)2?2
h4±16 564
h4±724
h4±6.084
h2 1.523.52 h2-1.520.48Therefore, the possible centers are at (6, 0) and (—2, 0).
Step 3: Write the Equations of the Circles
Using the general form of the circle equation (x - a)2 (y - b)2 r2, where a and b are the coordinates of the center and r is the radius:
For center (6, 0):(x-6)2 y225
x2-12x 36 y225
x2 y2-12x-110
For center (—2, 0):(x 2)2 y225
x2 4x 4 y225
x2 y2 4x-210
Thus, the equations of the circles are:
Final Equations
x2 y2-12x-110 x2 y2 4x-210Additional Notes: These are the final equations of the circles passing through the point (2, 3) with the center on the x-axis at coordinates (6, 0) and (—2, 0), respectively, and having a radius of 5 units.