You are drafting a circle on graph paper. The equation for the circle is x2+y2+12x−10y=−36 . Find the center and the radius of the circle.

Question

(1 point)
Responses

center: (12,−10), radius: 6 center: open paren 12 comma negative 10 close paren, radius: 6

center: (6,−5), radius: 6 center: open paren 6 comma negative 5 close paren, radius: 6

center: (−6,5), radius: 5center: open paren negative 6 comma 5 close paren, radius: 5

center: (−12,10), radius: 5

Answer 1

To find the center and radius of the circle defined by the equation \( x^2 + y^2 + 12x - 10y = -36 \), we need to rewrite this equation in standard form, which is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Starting from the given equation:

\[ x^2 + y^2 + 12x - 10y = -36 \]

1. Rearranging to make it easier to complete the square:

\[ x^2 + 12x + y^2 - 10y = -36 \]

2. Completing the square for the \(x\) terms:

   • \( x^2 + 12x \) can be completed as follows:
     • Take \(12\), halve it to get \(6\), and square it to get \(36\).
     • So, \( x^2 + 12x = (x + 6)^2 - 36 \).

3. Completing the square for the \(y\) terms:

   • \( y^2 - 10y \) can be completed similarly:
     • Take \(-10\), halve it to get \(-5\), and square it to get \(25\).
     • So, \( y^2 - 10y = (y - 5)^2 - 25 \).

4. Now substitute back into the equation:

   \[ (x + 6)^2 - 36 + (y - 5)^2 - 25 = -36 \]

   Simplifying the left side, we get:

   \[ (x + 6)^2 + (y - 5)^2 - 61 = -36 \]

5. Move \( -61 \) to the other side:

   \[ (x + 6)^2 + (y - 5)^2 = 25 \]

Now, we can identify the center and the radius:

• The center \((h, k)\) is \((-6, 5)\).
• The radius \(r\) is \(\sqrt{25} = 5\).

Thus, the center is \((-6, 5)\) and the radius is \(5\).

The correct answer is:

center: (−6, 5), radius: 5.