Question
Calculate the radius of the circle by completing the square of the equation x2+y2−14x−4y=11
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Answers
GPT 3.5
To calculate the radius of the circle, we need to complete the square of the given equation:
x^2 + y^2 - 14x - 4y = 11
Rearranging terms, we get:
(x^2 - 14x) + (y^2 - 4y) = 11
To complete the square for x, we take half of the coefficient of x (-14) and square it:
(-14/2)^2 = (-7)^2 = 49
Similarly, for y:
(-4/2)^2 = (-2)^2 = 4
Now we add the square terms to both sides of the equation:
(x^2 - 14x + 49) + (y^2 - 4y + 4) = 11 + 49 + 4
(x - 7)^2 + (y - 2)^2 = 64
Comparing this with the standard form of a circle equation:
(x - h)^2 + (y - k)^2 = r^2
We can see that the radius squared is 64, so the radius of the circle is:
r = √64 = 8
Therefore, the radius of the circle is 8 units.
x^2 + y^2 - 14x - 4y = 11
Rearranging terms, we get:
(x^2 - 14x) + (y^2 - 4y) = 11
To complete the square for x, we take half of the coefficient of x (-14) and square it:
(-14/2)^2 = (-7)^2 = 49
Similarly, for y:
(-4/2)^2 = (-2)^2 = 4
Now we add the square terms to both sides of the equation:
(x^2 - 14x + 49) + (y^2 - 4y + 4) = 11 + 49 + 4
(x - 7)^2 + (y - 2)^2 = 64
Comparing this with the standard form of a circle equation:
(x - h)^2 + (y - k)^2 = r^2
We can see that the radius squared is 64, so the radius of the circle is:
r = √64 = 8
Therefore, the radius of the circle is 8 units.