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Chapter 2: Problem 38

Find the center \((h, k)\) and radius \(r\) of each circle; \((b)\) graph each circle; \((c)\) find the intercepts, if any. $$ 3 x^{2}+3 y^{2}-12 y=0 $$

Short Answer

Expert verified
Center: \((0, 2)\), Radius: 2, Intercepts: \((0,0)\), \((0, 4)\)

Step by step solution

01

- Divide by the coefficient of squared terms

Given the equation: \[ 3x^2 + 3y^2 - 12y = 0 \]Divide every term by 3 to simplify: \[ x^2 + y^2 - 4y = 0 \]
02

- Complete the square

To complete the square for the y-terms, rewrite the equation as: \[ x^2 + (y^2 - 4y) = 0 \]Take half of the coefficient of y, square it, and add and subtract it inside the equation:\[ x^2 + (y^2 - 4y + 4 - 4) = 0 \]This becomes: \[ x^2 + (y - 2)^2 - 4 = 0 \]Add 4 to both sides to isolate the perfect square: \[ x^2 + (y-2)^2 = 4 \]
03

- Identify the circle's center and radius

The standard equation of a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \]Comparing this with \[ x^2 + (y - 2)^2 = 4 \], we see that the center is \((h, k) = (0, 2)\) and the radius is \(r\) such that \(r^2 = 4\), hence \(r = 2\).
04

- Graph the circle

To graph the circle, plot the center at \((0, 2)\), then use the radius of 2 units to draw a circle around the center, extending 2 units in all directions (up, down, left, right).
05

- Find the intercepts

To find the intercepts, first find the x-intercepts by setting \(y = 0\) and solving for \(x\):\[ x^2 + (0 - 2)^2 = 4 \]\therefore:\[ x^2 + 4 = 4 \]\[ x^2 = 0 \]\[ x = 0 \]So the x-intercept is \((0, 0)\). To find the y-intercepts, set \(x = 0\):\[ (0)^2 + (y - 2)^2 = 4 \]\therefore:\[ (y - 2)^2 = 4 \]\[ y - 2 = 卤2 \]\[ y = 4 \ or\ y = 0 \]So, the y-intercepts are \((0, 4)\) and \((0, 0)\).

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Key Concepts

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

completing the square
Completing the square is a crucial technique for simplifying quadratic equations, especially when dealing with circles. The idea is to transform a quadratic expression into a perfect square trinomial. This makes it easier to identify the center and radius of a circle.
To complete the square for an equation like \( x^2 + y^2 - 4y = 0 \), follow these steps:
  • Focus on the terms involving \( y \). Here, it's \( y^2 - 4y \).
  • Take half of the coefficient of \( y \), which is \( -4 \), divide by 2 to get \( -2 \), and then square it to get \( 4 \).
  • Add and subtract this square inside the equation, turning \( y^2 - 4y \) into \( (y - 2)^2 - 4 \).
This transformation allows you to rewrite the original equation more simply:
\ x^2 + (y - 2)^2 - 4 = 0 \.
By isolating the perfect square, you make it easy to convert the equation into the standard form of a circle.
center and radius of a circle
Understanding the center and radius of a circle is critical. The standard form of a circle equation is:
\ (x - h)^2 + (y - k)^2 = r^2 \.
Here, \( (h, k) \) represents the center, and \( r \) represents the radius.
From our simplified equation \ x^2 + (y - 2)^2 = 4 \, we can identify:
  • Center \( (h, k) = (0, 2) \)
  • Radius \( r \) as \( 2 \, because \ r^2 = 4 \).
To gather these values, compare your equation to the standard form and extract the coefficients. These values allow you to graph the circle accurately and understand its size and position.
intercepts of a circle
Finding intercepts involves determining where the circle crosses the x-axis and y-axis. To find intercepts, set the opposite variable to zero and solve the resulting equation.
For x-intercepts, set \ y = 0 \: \ x^2 + (0 - 2)^2 = 4 \
Simplify to find:
\ x^2 + 4 = 4 \ \Rightarrow x^2 = 0 \ \Rightarrow x = 0 \
So, the x-intercept is \( (0, 0) \).
For y-intercepts, set \ x = 0 \: \ (0)^2 + (y - 2)^2 = 4 \
This simplifies to:
\ (y - 2)^2 = 4 \ \Rightarrow y - 2 = \pm 2 \Rightarrow y = 0 \ or \ 4 \
Thus, the y-intercepts are \( (0, 0) \) and \( (0, 4) \).
Understanding intercepts is essential for graphing and analyzing the behavior of a circle on a coordinate plane.

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Most popular questions from this chapter

Cost Equation The fixed costs of operating a business are the costs incurred regardless of the level of production. Fixed costs include rent, fixed salaries, and costs of leasing machinery. The variable costs of operating a business are the costs that change with the level of output. Variable costs include raw materials, hourly wages, and electricity. Suppose that a manufacturer of jeans has fixed daily costs of \(\$ 1200\) and variable costs of \(\$ 20\) for each pair of jeans manufactured. Write a linear equation that relates the daily cost \(C,\) in dollars, of manufacturing the jeans to the number \(x\) of jeans manufactured. What is the cost of manufacturing 400 pairs of jeans? 740 pairs?

The volume \(V\) of a gas held at a constant temperature in a closed container varies inversely with its pressure \(P .\) If the volume of a gas is 600 cubic centimeters \(\left(\mathrm{cm}^{3}\right)\) when the pressure is 150 millimeters of mercury (mm \(\mathrm{Hg}\) ), find the volume when the pressure is \(200 \mathrm{~mm} \mathrm{Hg}\).

What can you say about two lines with the same \(x\) -intercept and the same \(y\) -intercept? Assume that the \(x\) -intercept is not \(0 .\)

Solve each equation. \(6 x-2(x-4)=24\)

(a) find the intercepts of the graph of each equation and (b) graph the equation. $$ -0.3 x+0.4 y=1.2 $$
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