Math  /  Geometry

QuestionCircle A in the xyx y-plane has the equation (x+5)2+(y5)2=4(x+5)^{2}+(y-5)^{2}=4. Circle B has the same center as circle A. The radius of circle B is two times the radius of circle AA. The equation defining circle B in the xyx y plane is (x+5)2+(y5)2=k(x+5)^{2}+(y-5)^{2}=k, where kk is a constant. What is the value of kk ?

Studdy Solution

STEP 1

What is this asking? We've got two circles, A and B, sharing the same center, but B is bigger!
We need to figure out just how much bigger and find the constant *k* that defines circle B's equation. Watch out! Don't mix up the radius and the *square* of the radius in the circle equation!

STEP 2

1. Find the radius of Circle A
2. Find the radius of Circle B
3. Determine *k*

STEP 3

The equation of a circle is (xh)2+(yk)2=r2(x-h)^2 + (y-k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius.
Circle A's equation is (x+5)2+(y5)2=4(x+5)^2 + (y-5)^2 = 4.

STEP 4

We see that r2=4r^2 = 4.
To find the radius rr of circle A, we take the square root of both sides: r2=4\sqrt{r^2} = \sqrt{4}, so r=2r = 2.
The radius of circle A is **2**!

STEP 5

The problem tells us the radius of circle B is *two times* the radius of circle A.

STEP 6

We just found that the radius of circle A is **2**.
So, the radius of circle B is 22=42 \cdot 2 = 4.

STEP 7

We know circle B has the same center as circle A, which is (5,5)(-5, 5) from the equation of circle A.

STEP 8

We also know the radius of circle B is **4**.
The equation of circle B is (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center and rr is the radius.

STEP 9

Plugging in the center (5,5)(-5, 5) and radius 44, we get (x+5)2+(y5)2=42(x+5)^2 + (y-5)^2 = 4^2, which simplifies to (x+5)2+(y5)2=16(x+5)^2 + (y-5)^2 = 16.

STEP 10

The problem tells us that the equation of circle B is (x+5)2+(y5)2=k(x+5)^2 + (y-5)^2 = k.
Comparing this with our equation, we see that k=16k = 16.

STEP 11

The value of *k* is **16**.

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