The value of k for which two tangents can be drawn from (k,k) to the circle `x^2+y^2+2x+2 y-16=0`

Options

A. `k in R^+`
B. `k in R^-`
C. `k in (- oo ,-4) uu (2,oo)`
D. `k in (0,1)`

Answer

To find the value of \( k \) for which two tangents can be drawn from the point \( (k, k) \) to the circle given by the equation \( x^2 + y^2 + 2x + 2y - 16 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation of the circle
The standard form of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \). We need to rewrite the given equation in this form.

Starting with:
\[ x^2 + y^2 + 2x + 2y - 16 = 0 \]
We can rearrange it as:
\[ x^2 + 2x + y^2 + 2y = 16 \]
Now, we complete the square for \( x \) and \( y \).

For \( x^2 + 2x \):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
For \( y^2 + 2y \):
\[ y^2 + 2y = (y + 1)^2 - 1 \]
Substituting back, we have:
\[ (x + 1)^2 - 1 + (y + 1)^2 - 1 = 16 \]
This simplifies to:
\[ (x + 1)^2 + (y + 1)^2 = 18 \]
Thus, the center of the circle is \( (-1, -1) \) and the radius \( r = \sqrt{18} = 3\sqrt{2} \).

### Step 2: Determine the condition for tangents
For two tangents to be drawn from the point \( (k, k) \) to the circle, the point must lie outside the circle. This means the distance from the point \( (k, k) \) to the center of the circle \( (-1, -1) \) must be greater than the radius of the circle.

### Step 3: Calculate the distance
The distance \( d \) from the point \( (k, k) \) to the center \( (-1, -1) \) is given by:
\[ d = \sqrt{(k - (-1))^2 + (k - (-1))^2} = \sqrt{(k + 1)^2 + (k + 1)^2} = \sqrt{2(k + 1)^2} = \sqrt{2} |k + 1| \]

### Step 4: Set up the inequality
We need the distance to be greater than the radius:
\[ \sqrt{2} |k + 1| > 3\sqrt{2} \]
Dividing both sides by \( \sqrt{2} \):
\[ |k + 1| > 3 \]

### Step 5: Solve the absolute value inequality
This gives us two inequalities:
1. \( k + 1 > 3 \) which simplifies to \( k > 2 \)
2. \( k + 1 < -3 \) which simplifies to \( k < -4 \)

### Step 6: Combine the results
Thus, the values of \( k \) for which two tangents can be drawn from the point \( (k, k) \) to the circle are:
\[ k < -4 \quad \text{or} \quad k > 2 \]

### Final Answer
The value of \( k \) for which two tangents can be drawn from the point \( (k, k) \) to the circle is:
\[ k \in (-\infty, -4) \cup (2, \infty) \]