4x^2+4y^2-16x-24y+51=0 is the equation of a circle. where is the center of the circle and what is it's radius?
A. center (2,3) radius=1/4
B. center (2,3) radius=1/2
C. center (-2,-3) radius=1/4
D. center (-2,-3) radius=1/2
It's B
To find the center and radius of a circle given its equation, we need to rewrite the equation in the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.
Let's start by completing the square for both x and y terms.
4x^2 - 16x + 4y^2 - 24y + 51 = 0
Rearranging the equation, we separate the x and y terms:
(4x^2 - 16x) + (4y^2 - 24y) = -51
Factor out the common factor of 4 from the x terms and the y terms:
4(x^2 - 4x) + 4(y^2 - 6y) = -51
Now, we need to add and subtract a constant term to complete the square for both x and y terms.
For the x terms:
We take half of the coefficient of x (-4) and square it (-4/2)^2 = 4. So we add and subtract 4 inside the parentheses.
For the y terms:
We take half of the coefficient of y (-6) and square it (-6/2)^2 = 9. So we add and subtract 9 inside the parentheses.
4(x^2 - 4x + 4) + 4(y^2 - 6y + 9) = -51 + 4(4) + 4(9)
Simplifying:
4(x - 2)^2 + 4(y - 3)^2 = -51 + 16 + 36
4(x - 2)^2 + 4(y - 3)^2 = 1
Dividing the equation by 4 to isolate the squared terms:
(x - 2)^2 + (y - 3)^2 = 1/4
Comparing this equation with the standard form, we can conclude that the center of the circle is (2, 3), and the radius is the square root of 1/4, which is 1/2. So, the correct answer is:
B. center (2,3) radius=1/2
To find the center and radius of a circle given its equation, we need to rewrite the equation in the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.
Let's rearrange the given equation:
4x^2 + 4y^2 - 16x - 24y + 51 = 0
First, group the x-terms and y-terms separately:
(4x^2 - 16x) + (4y^2 - 24y) = -51
Next, complete the square for both the x-terms and y-terms separately:
4(x^2 - 4x + 4) + 4(y^2 - 6y + 9) = -51 + 4(4) + 4(9)
Simplify the equation:
4(x - 2)^2 + 4(y - 3)^2 = -51 + 16 + 36
4(x - 2)^2 + 4(y - 3)^2 = 1
Divide both sides of the equation by 4 to get it into the standard form:
(x - 2)^2 + (y - 3)^2 = 1/4
Now we can identify the center of the circle as (2, 3) and the radius as the square root of the number on the right side, which is 1/2.
Therefore, the correct answer is option B: center (2, 3) radius = 1/2.