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Chapter 6: Problem 30

\(16 x^{2}+16 y^{2}-64 x+32 y+55=0\)

Short Answer

Expert verified
The center of the circle is at (2,-1) and the radius is 1.25.

Step by step solution

01

Rewriting the equation

First rewrite the equation by gathering the \(x\) terms and \(y\) terms: \(16(x^2 - 4x) +16(y^2 + 2y) = -55\)
02

Completing the square

Complete the square for both \(x\) and \(y\) terms: \(16[(x-2)^2 - 4] +16[(y+1)^2 -1] = -55\). Which simplifies to \(16(x-2)^2 +16(y+1)^2 = -55 + 64 + 16 = 25\)
03

Determine the center and radius

Divide the entire equation by 16 to isolate the squared terms on one side, which gives us \((x-2)^2 +(y+1)^2 = 1.5625\). For equations of the form \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is at (h,k) and the radius is \(r\). Therefore the center of this circle is at (2,-1) and the radius is \(\sqrt{1.5625} = 1.25\).

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Key Concepts

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

Equation of a Circle
The equation of a circle is a fundamental concept in geometry that describes the set of points which are equidistant from a given point, known as the center. In the standard form, this equation is represented as \[(x-h)^2 + (y-k)^2 = r^2\]Here,
  • \( (h, k) \) represent the coordinates of the center of the circle.
  • \( r \) is the radius of the circle.
This standard form is incredibly useful when you want to quickly identify the center and radius of a circle just by looking at the equation. When the terms in the equation involve squared expressions, the equation describes a circle with symmetric properties about its center.Understanding this form is crucial for identifying and drawing circles on a coordinate plane, making it a significant concept in coordinate geometry.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This modern approach connects algebra with geometry and allows for the solution of geometric problems using equations. In this framework,
  • points are defined by coordinates \((x, y)\)
  • the position of these points is determined numerically by their distance from axes
This powerful approach lets us analyze geometric shapes algebraically. For instance, determining the size, position, and properties of circles can be done using equations derived from the elements of coordinate geometry. By plotting the equation \[(x-h)^2 + (y-k)^2 = r^2\]on a coordinate plane, you can visualize the circle, making this relationship between equations and geometric figures very valuable in mathematical problem-solving.
Finding Center and Radius
When given a circle's equation in expanded form, such as \[16x^2 + 16y^2 - 64x + 32y + 55 = 0\],determining the center and radius involves a method known as completing the square. This process helps to transform the equation into the standard circle equation format.The steps are as follows:
  • Group and rearrange the terms: Gather the \(x\) and \(y\) terms together.
  • Apply completing the square: This involves rewriting the quadratic expressions in both \(x\) and \(y\) such that they form perfect squares.
  • Simplify and solve: Adjust the equation to match the standard form \((x-h)^2 + (y-k)^2 = r^2\).
For the exercise above, after simplifying, the equation becomes\[(x-2)^2 + (y+1)^2 = 1.5625\].From here, the center is identified as \((2, -1)\) and the radius as \(\sqrt{1.5625} = 1.25\). This transformation and identification are critical in understanding the geometric representation of the circle.

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Most popular questions from this chapter

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (4, \pi / 2),(1, \pi / 2) \end{array} $$

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi\). Show that the equation of the rotated graph is \(r=f(\theta-\phi) .\)

\(r=4+3 \cos \theta\)

In Exercises 83 and 84 , find the exact values of \(\sin 2 u\), \(\cos 2 u\), and \(\tan 2 u\) using the double-angle formulas. $$ \sin u=\frac{4}{5}, \frac{\pi}{2}

In Exercises 47-52, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. \(r=3-4 \cos \theta\)
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