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
B. Center (2,3) with a radius of 1/2
To determine the center and radius of a circle given its equation in standard form, we need to rewrite the equation in the form (x-h)^2 + (y-k)^2 = r^2, where (h,k) represents the center 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 the 'y' terms:
(4x^2 - 16x) + (4y^2 - 24y) + 51 = 0
Next, complete the square for the 'x' terms:
4(x^2 - 4x) + (4y^2 - 24y) + 51 = 0
To complete the square for x, we take half of the coefficient of x (-4), square it (4), and add it to both sides:
4(x^2 - 4x + 4) + (4y^2 - 24y) + 51 = 4(4)
Simplify:
4(x - 2)^2 + (4y^2 - 24y) + 51 = 16
Complete the square for the 'y' terms:
4(x - 2)^2 + 4(y^2 - 6y) + 51 = 16
To complete the square for y, we take half of the coefficient of y (-6), square it (9), and add it to both sides:
4(x - 2)^2 + 4(y^2 - 6y + 9) + 51 = 16 + 4(9)
Simplify:
4(x - 2)^2 + 4(y - 3)^2 + 51 = 52
Subtract 51 from both sides:
4(x - 2)^2 + 4(y - 3)^2 = 52 - 51
Simplify:
4(x - 2)^2 + 4(y - 3)^2 = 1
Now the equation is in the desired form: (x - h)^2 + (y - k)^2 = r^2
Comparing the given equation to the standard form, we can see that the center is (2,3) and the radius is 1/2.
Therefore, the correct answer is choice B. center (2,3) radius = 1/2.