Question

Determine the equation of a circle with center at $(-6,2)$ that is tangent to the $x$-axis (that is, the circle touches the $x$-axis at only one point).

Name: Problem 111, Chapter 10: Intermediate Algebra for College Students
Uploaded: 2020-08-05T18:57:40-08:00
Duration: 44 s
Channel: James Kiss
Description: Determine the equation of a circle with center at (-6,2) that is tangent to the x-axis (that is, the circle touches the x-axis at only one point).

   Determine the equation of a circle with center at $(-6,2)$ that is tangent to the $x$-axis (that is, the circle touches the $x$-axis at only one point).

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 10, Problem 111

Instant Answer

Solved by Expert James Kiss

08/05/2020

Step 1

Since the circle is tangent to the x-axis, the distance from the center of the circle to the x-axis is equal to the radius of the circle. The center of the circle is at (-6, 2), so the distance from the center to the x-axis is 2 units. Therefore, the radius of the Show more…

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Determine the equation of a circle with center at $(-6,2)$ that is tangent to the $x$-axis (that is, the circle touches the $x$-axis at only one point).

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Key Concepts

Distance from a Point to a Line

The formula for the distance from a point to a line is used to find how far the center of the circle is from the line it is tangent to. For horizontal lines like the x-axis, this distance is simply the absolute value of the y-coordinate of the center. This allows for easy computation of the circle’s radius in such scenarios.

Tangency Condition for a Circle

A circle is tangent to a line if it touches the line at exactly one point. In such cases, the perpendicular distance from the center of the circle to the line is equal to the radius. This principle helps in determining the radius when the condition of tangency is specified.

Standard Equation of a Circle

The standard equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the center and r is the radius. This form is fundamental in analytic geometry for defining all circles in the coordinate plane.

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