How to Find the Equation of a Concentric Circle with Given Circle and Tangent

Understanding the relationship between circles and tangents is a fundamental concept in geometry and beyond. When given the equation of a circle and a tangent, finding the equation of a concentric circle can be a rewarding exercise. This article will guide you through the process, providing a step-by-step methodology and an example to illustrate the procedure.

Introduction to Circle and Tangent

A circle is a geometric shape with all points on its boundary equidistant from the center. The general equation of a circle is ((x - h)^2 (y - k)^2 r^2), where ((h, k)) is the center and (r) is the radius. A tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at that point. The equation of the tangent line can be represented as (Ax By C 0).

Step-by-Step Process

Follow the below steps to find the equation of a concentric circle when you are provided with the equation of a circle and a tangent line:

Identify the Center and Radius of the Given Circle: The equation of the circle is in the form ((x - h)^2 (y - k)^2 r^2). The center of the circle is ((h, k)). The radius (r) is the square root of the right-hand side of the equation. Calculate the Distance from the Center to the Tangent Line: The formula for the distance (d) from a point ((h, k)) to the line (Ax By C 0) is given by: (d frac{|Ah Bk C|}{sqrt{A^2 B^2}}) Note the absolute value in the formula to ensure a non-negative result. Determine the New Radius of the Concentric Circle: The new radius for the concentric circle will be (r pm d). If the new circle is outside the original circle, the radius is (r d). If the new circle is inside the original circle, the radius is (r - d). Write the Equation of the Concentric Circle: Use the equation ((x - h)^2 (y - k)^2 (r pm d)^2). Substitute the values of (h), (k), and (r pm d) accordingly.

Example

Consider the given circle ((x - 3)^2 (y - 4)^2 16), with center ((3, 4)) and radius (4).

Identify the Center and Radius: Center ((h, k) (3, 4)) Radius (r 4) Tangent Line: Tangent line equation: (2x 3y - 12 0) (A 2), (B 3), (C -12) Calculate the Distance from the Center to the Tangent Line: (d frac{|2(3) 3(4) - 12|}{sqrt{2^2 3^2}} frac{|6 12 - 12|}{sqrt{4 9}} frac{6}{sqrt{13}}) Determine the New Radius: New radius ((r pm d)) For the concentric circle outside the tangent: (r d 4 frac{6}{sqrt{13}}) For the concentric circle inside the tangent: (r - d 4 - frac{6}{sqrt{13}}) Write the Equation of the Concentric Circle: For the circle outside the tangent: ((x - 3)^2 (y - 4)^2 left(4 frac{6}{sqrt{13}}right)^2) For the circle inside the tangent: ((x - 3)^2 (y - 4)^2 left(4 - frac{6}{sqrt{13}}right)^2)

Conclusion

Determining the equation of a concentric circle when provided with the equation of a circle and a tangent line involves identifying the center and radius of the circle, calculating the distance from the center to the tangent, and then adjusting the radius accordingly. This process is a valuable exercise in geometric reasoning and can be applied in various fields, including computer graphics and engineering.

Frequently Asked Questions (FAQ)

What is a concentric circle? A concentric circle is a circle that shares the same center as another circle but has a different radius. How do you find the equation of a circle from its center and radius? The general equation of a circle is ((x - h)^2 (y - k)^2 r^2), where ((h, k)) is the center and (r) is the radius. What does the term 'tangent' mean in the context of a circle? A tangent to a circle is a straight line that touches the circle at exactly one point and is perpendicular to the radius at that point.