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The coordinates for the center of the circle and the length of the radius are ( -7, -1 ) and 6 respectively.
Option D) is the correct answer.
Given the equation;
x² + y² + 14x + 2y + 14 = 0
We subtract 14 from both sides
x² + y² + 14x + 2y + 14 - 14 = 0 -14
x² + y² + 14x + 2y = -14
Next, we complete the square for x² + 14
We have, ( x + 7 )² - 49
Next, we complete the square for y² + 2y
We have, ( y + 1)² - 1
We substitute these squares into the given equation
x² + y² + 14x + 2y = -14
( x + 7 )² - 49 + ( y + 1)² - 1 = -14
Collect like terms
( x + 7 )² + ( y + 1)² = -14 + 49 + 1
( x + 7 )² + ( y + 1)² = 36
( x + 7 )² + ( y + 1)² = 6²
Note that, the form of a circle is given as;
( x-h )² + ( y-k)² = r²
Hence, we match the values into the form and determine our center and radius.
( x-(+7) )² + ( y-(+1))² = 6²
( x-7 )² + ( y-1 )² = 6²
Hence Center = ( -7, -1 ) and radius = 6
The coordinates for the center of the circle and the length of the radius are ( -7, -1 ) and 6 respectively.
Option D) is the correct answer.
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