Equation of a Circle Tangent to a Line: Standard Form and Calculation
When dealing with geometric shapes such as circles and lines, it’s important to understand how to derive the equation of a circle given certain conditions. Specifically, if a circle is tangent to a line, the distance from the center of the circle to this line is equivalent to the circle's radius. This article will guide you through the process of finding the standard form of the equation of a circle with a given center and tangent to a specific line.
Understanding the Problem
Consider a circle with its center at point ((-2, 3)) and that this circle is tangent to the line (y 1). The goal is to find the standard form of the circle's equation. The standard form of a circle's equation is:
Standard Form of a Circle's Equation
Step 1: Identify the Center and Line
The center of the circle is ((-2, 3)) and it is tangent to the line (y 1).
Step 2: Calculate the Radius
The radius of the circle is the distance from the center to the tangent line. We calculate this distance as the difference in the y-coordinates of the center and the line:
[text{Distance} |y_1 - y_2| |3 - 1| 2]
This distance is the radius of the circle, so (r 2).
Step 3: Write the Standard Form of the Circle's Equation
The standard form of the equation of a circle with center ((h, k)) and radius (r) is:
[ (x - h)^2 (y - k)^2 r^2 ]
Substituting (h -2), (k 3), and (r 2), we get:
[ (x 2)^2 (y - 3)^2 2^2 ]
Simplifying this, we arrive at the standard form of the circle's equation:
[ (x 2)^2 (y - 3)^2 4 ]
Length of Perpendicular from a Point to a Line
Another way to find the radius involves using the perpendicular distance from a point to a line. The length of the perpendicular from the point ((-2, 3)) to the line (y 1) (which is a horizontal line) is simply the absolute value of the difference between the y-coordinates:
[text{Length} |3 - 1| 2]
Thus, the radius of the circle is 2, and the standard form of the circle's equation is:
[ (x 2)^2 (y - 3)^2 4 ]
General Form to Standard Form
To convert the standard form to the general form of the circle's equation, expand the squared binomials:
[ (x 2)^2 (y - 3)^2 4 ]
[ x^2 4x 4 y^2 - 6y 9 4 ]
[ x^2 y^2 4x - 6y 13 4 ]
[ x^2 y^2 4x - 6y 9 0 ]
Conclusion
Understanding the relationship between a circle and the line it is tangent to is crucial for solving geometric problems. By leveraging the distance formula and the standard form of the circle's equation, we can easily determine the equation of a circle given its center and a tangent line. This knowledge can be extended to other conic sections such as hyperbolas and parabolas as well, where completing the square is a fundamental technique.