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
hint:
complete the square
To determine the center and radius of a circle given its equation, we need to rewrite it in a specific format. The standard form for the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle, and r represents the radius.
Let's manipulate the given equation to match this form.
Starting with: 4x^2 + 4y^2 - 16x - 24y + 51 = 0
Step 1: Group the x-terms and y-terms separately:
(4x^2 - 16x) + (4y^2 - 24y) + 51 = 0
Step 2: Factor out the common coefficients for the x-terms and y-terms:
4(x^2 - 4x) + 4(y^2 - 6y) + 51 = 0
Step 3: Complete the square for x and y separately:
4(x^2 - 4x + 4) + 4(y^2 - 6y + 9) - 16 + 36 + 51 = 0
Simplifying:
4(x - 2)^2 + 4(y - 3)^2 + 71 = 0
Rearranging terms:
4(x - 2)^2 + 4(y - 3)^2 = -71
Dividing both sides by 4:
(x - 2)^2 + (y - 3)^2 = -71/4
Now we can see that the equation is not in standard form since the right side is negative. This implies that the given equation does not represent a circle. Therefore, none of the answer choices (A, B, C, or D) are correct.
Please double-check the equation or provide the correct equation representing a circle to determine the center and radius accurately.