If the length of the tangent from (h,k) to the circle `x^(2)+y^2=16` is twice the length of the tangent from the same point to the circle `x^(2)+y^(2)+2x+2y=0`, then

Options

A. `h^(2)+k^(2)_4h+4k+16=0`
B. `h^(2)+k^(2)+3h+3k=0`
C. `3h^(2)+3k^(2)+8y+8k+16=0`
D. `3h^(2)+3k^(2)+4h+4k+16=0`

Answer

To solve the problem, we need to find the relationship between the point \((h, k)\) and the two circles given.

### Step-by-Step Solution:

1. **Identify the equations of the circles:**
- The first circle is given by the equation \(x^2 + y^2 = 16\).
- The second circle is given by the equation \(x^2 + y^2 + 2x + 2y = 0\). We can rewrite this as \(x^2 + y^2 + 2x + 2y + 4 = 4\), which simplifies to \((x + 1)^2 + (y + 1)^2 = 4\). This means the center of the second circle is at \((-1, -1)\) with a radius of \(2\).

2. **Calculate the lengths of the tangents:**
- The length of the tangent from a point \((h, k)\) to the first circle is given by the formula:
\[ L_1 = \sqrt{h^2 + k^2 - 16} \]
- The length of the tangent from the same point \((h, k)\) to the second circle is:
\[ L_2 = \sqrt{(h + 1)^2 + (k + 1)^2 - 4} \]

3. **Set up the relationship between the lengths:**
- According to the problem, the length of the tangent to the first circle is twice that of the second circle:
\[ L_1 = 2L_2 \]
- Substituting the expressions for \(L_1\) and \(L_2\):
\[ \sqrt{h^2 + k^2 - 16} = 2\sqrt{(h + 1)^2 + (k + 1)^2 - 4} \]

4. **Square both sides to eliminate the square roots:**
\[ h^2 + k^2 - 16 = 4\left((h + 1)^2 + (k + 1)^2 - 4\right) \]

5. **Expand the right-hand side:**
- Expanding \((h + 1)^2\) gives \(h^2 + 2h + 1\).
- Expanding \((k + 1)^2\) gives \(k^2 + 2k + 1\).
- Thus, we have:
\[ h^2 + k^2 - 16 = 4(h^2 + 2h + 1 + k^2 + 2k + 1 - 4) \]
- Simplifying the right-hand side:
\[ = 4(h^2 + k^2 + 2h + 2k - 2) = 4h^2 + 4k^2 + 8h + 8k - 8 \]

6. **Rearranging the equation:**
- Bringing all terms to one side:
\[ h^2 + k^2 - 16 - 4h^2 - 4k^2 - 8h - 8k + 8 = 0 \]
- This simplifies to:
\[ -3h^2 - 3k^2 - 8h - 8k - 8 = 0 \]
- Multiplying through by \(-1\):
\[ 3h^2 + 3k^2 + 8h + 8k + 8 = 0 \]

7. **Final equation:**
- Dividing through by 3 gives:
\[ h^2 + k^2 + \frac{8}{3}h + \frac{8}{3}k + \frac{8}{3} = 0 \]

### Conclusion:
The required equation is:
\[ 3h^2 + 3k^2 + 8h + 8k + 8 = 0 \]