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Chapter 11: Problem 6

The standard equation of a circle with center at (2,-3) and radius 1 is ______

Short Answer

Expert verified
(x - 2)^2 + (y + 3)^2 = 1

Step by step solution

01

Understand the Standard Equation of a Circle

The standard form of the equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \(h\) and \(k\) are the coordinates of the center and \(r\) is the radius.
02

Substitute the Center Coordinates

Given the center of the circle is \( (2, -3) \). Substitute \( h = 2 \) and \( k = -3 \) into the standard equation: \( (x - 2)^2 + (y + 3)^2 = r^2 \).
03

Insert the Radius

Given the radius of the circle is \( 1 \). Substitute \( r = 1 \) into the equation: \( (x - 2)^2 + (y + 3)^2 = 1^2 \).
04

Simplify the Equation

Simplify the equation: \( (x - 2)^2 + (y + 3)^2 = 1 \).

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Key Concepts

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

circle center
The center of a circle is an important concept in understanding and forming the circle's equation. It is represented by a point with coordinates \( \( h, k \) \), where \( h \) is the x-coordinate and \( k \) is the y-coordinate of the center. For example, if the center of a circle is at the point (2, -3), this means that the circle is positioned 2 units to the right of the origin and 3 units downward. Knowing the center is crucial because it helps us use the standard form of the circle's equation effectively.
To sum up:
  • The center is denoted as (h, k).
  • The coordinates dictate the circle鈥檚 position on the Cartesian plane.
circle radius
The radius of a circle is the distance from the center to any point on the circle. It is a constant value denoted by \( r \). In the formula, this radius is squared (\( r^2 \)) to relate it to the coordinates. For a radius of length 1, the point on the circle is exactly one unit away from the center, regardless of direction. This value helps determine how large or small the circle appears.
To recap:
  • The radius is the distance from the center to any point on the circle.
  • This distance remains the same at all points around the circle.
  • It is denoted by \( r \).
standard form of a circle equation
The standard form of a circle equation is essential for quickly identifying the key traits of a circle, namely, its center and radius. The general formula is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius.
Here's a step-by-step breakdown:
  • First, recognize that the formula helps relate the x and y coordinates of any point on the circle to the center.
  • By substituting the center (h, k) and squaring the radius (r2), you form a specific equation for that circle.
  • For example, with a center at (2, -3) and a radius of 1, the equation becomes \( (x - 2)^2 + (y + 3)^2 = 1 \). This follows directly from substituting 2 for h, -3 for k, and 1 for r.
    Therefore, the standard form provides a comprehensive summary of the circle's essential properties in a clean mathematical expression.

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Most popular questions from this chapter

A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center.

A rectangle is inscribed in an ellipse with major axis of length 14 meters and minor axis of length 4 meters. Find the maximum area of a rectangle inscribed in the ellipse. Round your answer to two decimal places.

Find an equation for the hyperbola described. Graph the equation. Vertices at (0,-6) and (0,6)\(;\) asymptote the line \(y=2 x\)

Use a graphing utility to graph the plane curve defined by the given parametric equations. \(x(t)=\sin t+\cos t, \quad y=\sin t-\cos t\)

Graph each function. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.] \(f(x)=\sqrt{16+4 x^{2}}\)
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