Question

center: (1, -5)
radius: 3

Explanation:

To determine the equation of the circle with center \((1, -5)\) and radius \(3\), we use the standard form of the equation of a circle:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 1: Identify the values of \(h\), \(k\), and \(r\)

• The center of the circle is \((1, -5)\), so \(h = 1\) and \(k = -5\).
• The radius \(r = 3\).

Step 2: Substitute the values into the standard form

Substitute \(h = 1\), \(k = -5\), and \(r = 3\) into the equation:

\[
(x - 1)^2 + (y - (-5))^2 = 3^2
\]

Simplify the equation:

\[
(x - 1)^2 + (y + 5)^2 = 9
\]

So, the equation of the circle is \((x - 1)^2 + (y + 5)^2 = 9\).

To graph the circle:

1. Locate the center \((1, -5)\) on the coordinate plane.
2. From the center, move 3 units up, down, left, and right to find four points on the circle:

• Up: \((1, -5 + 3) = (1, -2)\)
• Down: \((1, -5 - 3) = (1, -8)\)
• Left: \((1 - 3, -5) = (-2, -5)\)
• Right: \((1 + 3, -5) = (4, -5)\)

1. Plot these points and draw a smooth circle through them.

Answer:

To determine the equation of the circle with center \((1, -5)\) and radius \(3\), we use the standard form of the equation of a circle:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 1: Identify the values of \(h\), \(k\), and \(r\)

• The center of the circle is \((1, -5)\), so \(h = 1\) and \(k = -5\).
• The radius \(r = 3\).

Step 2: Substitute the values into the standard form

Substitute \(h = 1\), \(k = -5\), and \(r = 3\) into the equation:

\[
(x - 1)^2 + (y - (-5))^2 = 3^2
\]

Simplify the equation:

\[
(x - 1)^2 + (y + 5)^2 = 9
\]

So, the equation of the circle is \((x - 1)^2 + (y + 5)^2 = 9\).

To graph the circle:

1. Locate the center \((1, -5)\) on the coordinate plane.
2. From the center, move 3 units up, down, left, and right to find four points on the circle:

• Up: \((1, -5 + 3) = (1, -2)\)
• Down: \((1, -5 - 3) = (1, -8)\)
• Left: \((1 - 3, -5) = (-2, -5)\)
• Right: \((1 + 3, -5) = (4, -5)\)

1. Plot these points and draw a smooth circle through them.