Determining the Distance of a Circle from the X-axis: A Step-by-Step Guide
Understanding how far a circle is from the x-axis can be crucial in various mathematical applications. In this article, we will walk through the process of determining this distance for a specific circle, using the equation x2 y2 - 8x 1 0. We will explore the steps to rewrite the equation in standard form, identify the center and radius of the circle, and calculate the required distance.
1. Rewriting the Circle Equation in Standard Form
Given the circle equation x2 y2 - 8x 1 0, we need to rewrite it in the standard form (x-h)2 (y-k)2 r2.
First, we complete the square for the x terms. Take half of the coefficient of x, which is -8. Half of -8 is -4, and squaring it gives 16. Completing the square, we get: x2 - 8x 16 Substitute back into the original equation and simplify: x2 - 8x 16 y2 1 - 16 0 Further simplification gives: (x - 4)2 y2 - 15 0 Which can be rewritten as: (x - 4)2 y2 152. Identifying the Center and Radius of the Circle
The standard form of the circle's equation is now clear: (x - 4)2 (y - 0)2 15. From this, we identify:
Center: (4, 0) Radius: √153. Calculating the Distance from the Circle to the X-axis
To determine the distance from the circle to the x-axis, we need to consider the following:
The distance from the center of the circle to the x-axis is simply the y-coordinate of the center, which is 0. The radius of the circle is √15. The distance from the x-axis to the edge of the circle is the radius, √15.Therefore, the distance from the circle to the x-axis is √15, considering the radius of the circle.
4. Alternative Method: Completing the Square
We can also attempt to solve the equation by setting y 0 to find the x-intercepts:
Set y 0 in the original equation: x2 - 8x 1 0 Solve for x using the quadratic formula: x [8 ± √(64 - 4 * 1 * 1)] / 2 x [8 ± √60] / 2 x 4 ± √15 The x-intercepts are x 4 √15 and x 4 - √15. The distance from these points to the x-axis is 0, as they lie on the x-axis.Conclusion
By rewriting the given circle equation and identifying its center and radius, we can determine that the distance from this circle to the x-axis is √15. The alternative method of finding the x-intercepts shows that the circle indeed touches the x-axis at these points, further confirming our initial calculation.
Understanding how to manipulate and interpret circle equations is an essential skill in mathematics and can find applications in various fields such as physics, engineering, and computer graphics.