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Chapter 2: Problem 61

55鈥�62 ? Find an equation of the circle that satisfies the given conditions. Center \((7,-3) ; \quad\) tangent to the \(x\) -axis

Short Answer

Expert verified
The circle's equation is \((x-7)^2 + (y+3)^2 = 9\).

Step by step solution

01

Identify Circle Equation

The general form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\) where \((h, k)\) is the center and \(r\) is the radius. Here, the center is given as \((7, -3)\).
02

Determine the Radius

Since the circle is tangent to the x-axis, the distance from the center \((7, -3)\) to the x-axis represents the radius. This distance is the absolute value of the y-coordinate of the center, \(|-3| = 3\).
03

Write the Specific Circle Equation

Substitute the center \((h, k) = (7, -3)\) and radius \(r = 3\) into the circle equation. This yields: \((x-7)^2 + (y+3)^2 = 3^2\).
04

Simplify the Equation

Calculate the square of the radius: \(3^2 = 9\). Therefore, the equation of the circle becomes \((x-7)^2 + (y+3)^2 = 9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius of a Circle
The radius of a circle is a fundamental part of its geometry. It is the distance from the center of the circle to any point on its edge. This distance is constant for a given circle and determines its size. In mathematical terms, the radius, which we often denote by \(r\), plays a crucial role in the formula for the area and circumference of the circle.
  • The radius is always positive.
  • It is half of the diameter, which is the longest distance across the circle through its center.
  • The formula for the circumference of a circle is \(2\pi r\).
  • The formula for the area of a circle is \(\pi r^2\).
In this exercise, the circle is tangent to the x-axis. This means that the distance from the center of the circle to the x-axis is exactly equal to the radius. The center of the circle is given as \((7, -3)\), so the radius is the absolute value of the y-coordinate, \( |-3| = 3 \).
Center of a Circle
The center of a circle is defined as the point that is equidistant from any point on the circle's boundary. This point essentially acts as a reference from which all points on the circle are equidistant, and it is represented in the standard form of the circle's equation. The general equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center.
  • The center鈥檚 coordinates \((h, k)\) indicate how the circle is positioned on a coordinate plane.
  • A circle centered at the origin has the equation \(x^2 + y^2 = r^2\).
  • Any translation of the circle horizontally or vertically is reflected in the different values of \(h\) and \(k\).
In this exercise, the center of the circle is given as \((7, -3)\). This means that the circle is located 7 units to the right and 3 units down from the origin in the coordinate plane.
Tangent to the X-Axis
A circle that is tangent to the x-axis has one of its boundaries just touching the axis at a single point. This specific condition offers a piece of valuable information about the circle. It tells us a relationship between the radius and the position of the center. For the circle to be tangent to the x-axis, the vertical distance from the circle鈥檚 center to the x-axis has to be exactly equal to the radius.
  • The tangent point is the closest point on the circle to the x-axis.
  • A circle tangent to the x-axis at \((h, k)\) will have the y-coordinate of its center equal to either \(r\) or \(-r\).
  • This helps in quickly determining the radius if the center is known.
For the exercise at hand, since the center is at \((7, -3)\), the circle is tangent to the x-axis where \(y = 0\). Thus, the radius must be equal to the absolute value of the y-coordinate of the center, \(|-3| = 3\), as it extends upward to just touch the x-axis.

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Most popular questions from this chapter

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