Problem 24 - Algebra 2

Question

For each equation, find the center and radius of the circle. $$ (x+6)^{2}+y^{2}=121 $$

Step-by-Step Solution

Answer

The center of the circle is at (6, 0) and the radius is 11.

Step 1: Identify the Center Point

Firstly, we can identify the center of the circle by examining the equation. The center is given by the terms opposite the x and y coefficients. In our case, the x coefficient is -6 obtained from the term (x+6). Therefore, the x-coordinate of the center is -(-6) = 6. The y coefficient is not present which means it is 0, thus the y-coordinate of the center is -0 = 0. Hence, center of circle is at (6, 0).

Step 2: Identify the Radius

Secondly, we can identify the radius by finding the square root of the constant term on the right side of the equation, which in this case, is 121. The square root of 121 is 11. Therefore, the radius of the circle is 11.

Key Concepts

Center of a Circle

To find the center of a circle when you're given the equation in standard form, focus on the terms that involve variables squared. A circle's equation typically looks like this: \[(x-h)^2 + (y-k)^2 = r^2\] where

• \(h\)
• \(k\)

are the coordinates of the circle's center.The equation \[(x+6)^2 + y^2 = 121\] looks a bit different, but you can still spot the center.For the x-term, \((x+6)\),it can be rewritten as \((x - (-6))\). This tells us that the x-coordinate of the center is at \(-6\). Since there's no additional term next to \(y^2\), it indicates that the y-component comes from \(y - 0\), meaning the y-coordinate is \(0\). So, the center of the circle is \((-6, 0)\).Once you know this, you have automatically mapped out where the circle sits on the coordinate plane.

Radius of a Circle

Identifying the radius from a circle's equation is straightforward once it's in standard form. In the form \((x - h)^2 + (y - k)^2 = r^2\),\( r\) represents the radius of the circle.For our equation \[(x+6)^2 + y^2 = 121\], the number \(121\)on the right side is equal to \(r^2\).This means we need to find the square root of \(121\) to determine the radius.

• Finding the square root of \(121\) is simple.It results in \(11\).

Therefore, the radius of the circle is \(11\).Whenever you're given a constant on the right side of these equations, taking the square root provides the circle’s radius. This is vital because it tells us how large the circle is and how far its edge is from the center.

Equation of a Circle

The equation of a circle plays a central role in defining its properties and placement on the coordinate plane. It encapsulates two main factors: the center and the radius. The standard form for a circle’s equation is:\[(x - h)^2 + (y - k)^2 = r^2\]This form tells us:

• The coordinates \((h, k)\) which mark the center of the circle.
• The term \(r^2\) from which we can derive the radius.

To illustrate, let’s consider our example \((x+6)^2 + y^2 = 121\).By rearranging the terms, we identify the components that summarize the circle:- The \(x\)term’s structure \((x+6)transforming to (x - (-6))\)points to the center having an x-coordinate of \(-6\).- The lack of a y-term similar to \((y-k)\)means \(y\) is centered at \(0\).This highlights the circle's center at \((-6, 0)\),while- The right side, \(121, matches the form r^2\) indicating its radius is \(11\). Combining these insights gives us a circle centered at \((-6, 0)\), with a radius extending 11 units in all directions.