Which equation represents the circle described? The radius is 2 units. The center is the same as the center of a circle whose equation is x^2 + y^2 – 8x – 6y + 24 = 0. a. (x + 4)^2 + (y + 3)^2 = 2 b. (x – 4)^2 + (y – 3)^2 = 2 c. (x – 4)^2 + (y – 3)^2 = 2^2 d. (x + 4)^2 + (y + 3)^2 = 2^2
The Correct Answer and Explanation is:
To find the correct equation of the circle, follow these steps:
Step 1: Identify the center of the given circle
We are given a circle with the equation: x2+y2−8x−6y+24=0x^2 + y^2 – 8x – 6y + 24 = 0x2+y2−8x−6y+24=0
To find its center, complete the square.
Group the terms: (x2−8x)+(y2−6y)+24=0(x^2 – 8x) + (y^2 – 6y) + 24 = 0(x2−8x)+(y2−6y)+24=0
Complete the square for both groups:
- For x2−8xx^2 – 8xx2−8x, add and subtract 161616 (82)2(\frac{8}{2})^2(28)2
- For y2−6yy^2 – 6yy2−6y, add and subtract 999 (62)2(\frac{6}{2})^2(26)2
(x2−8x+16)+(y2−6y+9)+24−16−9=0(x^2 – 8x + 16) + (y^2 – 6y + 9) + 24 – 16 – 9 = 0(x2−8x+16)+(y2−6y+9)+24−16−9=0 (x−4)2+(y−3)2−1=0(x – 4)^2 + (y – 3)^2 – 1 = 0(x−4)2+(y−3)2−1=0 (x−4)2+(y−3)2=1(x – 4)^2 + (y – 3)^2 = 1(x−4)2+(y−3)2=1
So, the center of this circle is (4,3)(4, 3)(4,3)
Step 2: Use the new radius
Now, we are told to find a new circle with the same center (4,3)(4, 3)(4,3) and radius of 2 units
The standard form of a circle’s equation is: (x−h)2+(y−k)2=r2(x – h)^2 + (y – k)^2 = r^2(x−h)2+(y−k)2=r2
Where (h,k)(h, k)(h,k) is the center and rrr is the radius.
So the equation becomes: (x−4)2+(y−3)2=22(x – 4)^2 + (y – 3)^2 = 2^2(x−4)2+(y−3)2=22
Final Answer:
c. (x−4)2+(y−3)2=22(x – 4)^2 + (y – 3)^2 = 2^2(x−4)2+(y−3)2=22
Explanation:
This equation correctly reflects a circle with radius 2 and center (4,3)(4, 3)(4,3). It matches choice c, which shows the squared radius rather than the simplified number. This form is acceptable and commonly used to emphasize the actual radius. Choices a, b, and d either have the wrong center or do not correctly express the squared radius.
