Dedric is trying to graph a circle on a computer software program that has a grid. The equation he is graphing is x2+y2−20x−14y+5=0 . Where would he place the center of the circle and what is its radius?(1 point)

Question

Responses

center=(10,7), radius=12
center= left parenthesis 10 comma 7 right parenthesis , radius=12

center=(20,14), radius=5
center= left parenthesis 20 comma 14 right parenthesis , radius=5

center=(−20,−14), radius=8
center= left parenthesis negative 20 comma negative 14 right parenthesis , radius=8

center=(−10,−7), radius=5

Answer 1

To determine the center and radius of the circle given the equation \( x^2 + y^2 - 20x - 14y + 5 = 0 \), we first need to rewrite it in standard form.

1. Rearranging the equation: \[ x^2 - 20x + y^2 - 14y + 5 = 0 \] \[ x^2 - 20x + y^2 - 14y = -5 \]

2. Completing the square:

   • For \( x^2 - 20x \): \[ x^2 - 20x \rightarrow \left( x - 10 \right)^2 - 100 \]

   • For \( y^2 - 14y \): \[ y^2 - 14y \rightarrow \left( y - 7 \right)^2 - 49 \]

3. Substituting back into the equation: \[ \left( x - 10 \right)^2 - 100 + \left( y - 7 \right)^2 - 49 = -5 \] \[ \left( x - 10 \right)^2 + \left( y - 7 \right)^2 - 149 = -5 \] \[ \left( x - 10 \right)^2 + \left( y - 7 \right)^2 = 144 \]

4. Identifying the center and radius: The equation \( \left( x - 10 \right)^2 + \left( y - 7 \right)^2 = 12^2 \) indicates that the center of the circle is \( (10, 7) \) and the radius is \( 12 \).

The correct response is: center=(10,7), radius=12.