Question

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

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

Step-by-step solution

- Divide by the coefficient of squared terms

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

- 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: \[ x^2 + (y-2)^2 = 4 \]

- 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 \(r = 2\).

- 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).

- 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:\[ (y - 2)^2 = 4 \]\[ y - 2 = ±2 \]\[ y = 4 \ or\ y = 0 \]So, the y-intercepts are \((0, 4)\) and \((0, 0)\).

Key concepts

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.