If the length of tangent drawn from the point (5,3) to the circle `x^2+y^2+2x+ky+17=0` is 7, then k= ?

Options

A. -6
B. -4
C. `13/2`
D. 4

Answer

To solve the problem, we need to find the value of \( k \) given that the length of the tangent from the point \( (5, 3) \) to the circle described by the equation \( x^2 + y^2 + 2x + ky + 17 = 0 \) is 7.

### Step-by-Step Solution:

1. **Identify the Circle's Equation**:
The equation of the circle is given as:
\[ x^2 + y^2 + 2x + ky + 17 = 0 \]
We can rewrite this in the standard form \( (x + g)^2 + (y + f)^2 = r^2 \) by completing the square.

2. **Complete the Square**:
- For \( x^2 + 2x \):
\[ x^2 + 2x = (x + 1)^2 - 1 \]
- For \( y^2 + ky \):
\[ y^2 + ky = (y + \frac{k}{2})^2 - \left(\frac{k}{2}\right)^2 \]
- Substitute these back into the circle's equation:
\[ (x + 1)^2 - 1 + (y + \frac{k}{2})^2 - \left(\frac{k}{2}\right)^2 + 17 = 0 \]
- Simplifying gives:
\[ (x + 1)^2 + (y + \frac{k}{2})^2 = \left(\frac{k}{2}\right)^2 - 16 \]

3. **Identify Circle Parameters**:
From the standard form, we can identify:
- Center: \( (-1, -\frac{k}{2}) \)
- Radius: \( r = \sqrt{\left(\frac{k}{2}\right)^2 - 16} \)

4. **Length of Tangent Formula**:
The length \( L \) of the tangent from a point \( (x_1, y_1) \) to a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ L = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \]
Here, \( (x_1, y_1) = (5, 3) \) and \( (h, k) = (-1, -\frac{k}{2}) \).

5. **Substituting Values**:
Substitute into the formula:
\[ L = \sqrt{(5 - (-1))^2 + (3 - (-\frac{k}{2}))^2 - \left(\sqrt{\left(\frac{k}{2}\right)^2 - 16}\right)^2} \]
Simplifying gives:
\[ L = \sqrt{(6)^2 + \left(3 + \frac{k}{2}\right)^2 - \left(\frac{k}{2}\right)^2 + 16} \]

6. **Set Length Equal to 7**:
Since the length of the tangent is given as 7, we have:
\[ 7 = \sqrt{36 + \left(3 + \frac{k}{2}\right)^2 - \left(\frac{k}{2}\right)^2 + 16} \]
Squaring both sides:
\[ 49 = 36 + 16 + \left(3 + \frac{k}{2}\right)^2 - \left(\frac{k}{2}\right)^2 \]
Simplifying gives:
\[ 49 = 52 + 3k + \frac{k^2}{4} - \frac{k^2}{4} \]
Thus:
\[ 49 = 52 + 3k \]

7. **Solve for k**:
Rearranging gives:
\[ 3k = 49 - 52 \]
\[ 3k = -3 \]
\[ k = -1 \]

### Final Answer:
The value of \( k \) is \( -1 \).