The distance between the chords of contact of the tangents to the circle `x ^(2) + y ^(2) + 32 x + 24 y -1 =0` from the origin and the point (16,12) is k. The value of k is

Answer

To find the distance \( k \) between the chords of contact of the tangents to the circle \( x^2 + y^2 + 32x + 24y - 1 = 0 \) from the origin and the point \( (16, 12) \), we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 + 32x + 24y - 1 = 0 \]
We can complete the square for \( x \) and \( y \).

For \( x \):
\[ x^2 + 32x = (x + 16)^2 - 256 \]
For \( y \):
\[ y^2 + 24y = (y + 12)^2 - 144 \]
Substituting these back into the equation gives:
\[ (x + 16)^2 - 256 + (y + 12)^2 - 144 - 1 = 0 \]
\[ (x + 16)^2 + (y + 12)^2 = 401 \]
Thus, the center of the circle is \( (-16, -12) \) and the radius is \( \sqrt{401} \).

### Step 2: Find the Chord of Contact from the Origin
The formula for the chord of contact from a point \( (x_1, y_1) \) to the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) is given by:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
For the origin \( (0, 0) \):
- \( g = 16 \) (since \( 2g = 32 \))
- \( f = 12 \) (since \( 2f = 24 \))
- \( c = -1 \)

Substituting \( (x_1, y_1) = (0, 0) \):
\[ x \cdot 0 + y \cdot 0 + 16(x - 0) + 12(y - 0) - 1 = 0 \]
This simplifies to:
\[ 16x + 12y - 1 = 0 \]

### Step 3: Find the Chord of Contact from the Point (16, 12)
Now, we find the chord of contact from the point \( (16, 12) \):
Substituting \( (x_1, y_1) = (16, 12) \):
\[ x \cdot 16 + y \cdot 12 + 16(x - 16) + 12(y - 12) - 1 = 0 \]
This simplifies to:
\[ 16x + 12y + 16x - 256 + 12y - 144 - 1 = 0 \]
\[ 32x + 24y - 401 = 0 \]

### Step 4: Find the Distance Between the Two Parallel Lines
We have the two equations:
1. \( 16x + 12y - 1 = 0 \) (Equation 1)
2. \( 32x + 24y - 401 = 0 \) (Equation 2)

To find the distance \( k \) between these two parallel lines, we use the formula for the distance between two parallel lines:
\[ k = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]
Where \( c_1 = -1 \) and \( c_2 = -401 \) (after rearranging the second equation).

Calculating:
\[ k = \frac{|-401 - (-1)|}{\sqrt{16^2 + 12^2}} = \frac{|-400|}{\sqrt{256 + 144}} = \frac{400}{\sqrt{400}} = \frac{400}{20} = 20 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{20} \]