Find the chord of contact of `(2, 5)` with
repect ot the circle
`x^(2) + y^(2) - 5x + 4y-2=0`.

Answer

To find the chord of contact of the point \( (2, 5) \) with respect to the circle given by the equation

\[ x^2 + y^2 - 5x + 4y - 2 = 0, \]

we can follow these steps:

### Step 1: Identify the point and circle parameters
The point given is \( (x_1, y_1) = (2, 5) \). The equation of the circle can be compared with the standard form:

\[ x^2 + y^2 + 2gx + 2fy + c = 0. \]

From the given equation, we can identify the coefficients:
- \( 2g = -5 \) (hence \( g = -\frac{5}{2} \))
- \( 2f = 4 \) (hence \( f = 2 \))
- \( c = -2 \)

### Step 2: Write the chord of contact equation
The equation of the chord of contact from a point \( (x_1, y_1) \) with respect to the circle is given by:

\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0. \]

Substituting the values \( x_1 = 2 \), \( y_1 = 5 \), \( g = -\frac{5}{2} \), \( f = 2 \), and \( c = -2 \):

\[ x(2) + y(5) + \left(-\frac{5}{2}\right)(x + 2) + 2(y + 5) - 2 = 0. \]

### Step 3: Simplify the equation
Now, we simplify the equation:

1. Expand the terms:
\[ 2x + 5y - \frac{5}{2}x - 5 + 2y + 10 - 2 = 0. \]

2. Combine like terms:
\[ \left(2x - \frac{5}{2}x\right) + (5y + 2y) + (-5 + 10 - 2) = 0, \]
which simplifies to:
\[ \left(2 - \frac{5}{2}\right)x + 7y + 3 = 0. \]

3. Calculate \( 2 - \frac{5}{2} \):
\[ 2 = \frac{4}{2} \Rightarrow 2 - \frac{5}{2} = \frac{4 - 5}{2} = -\frac{1}{2}. \]

Thus, we have:
\[ -\frac{1}{2}x + 7y + 3 = 0. \]

### Step 4: Multiply through by -2 to eliminate the fraction
To make the equation neater, multiply through by -2:
\[ x - 14y - 6 = 0. \]

### Final Answer
The chord of contact of the point \( (2, 5) \) with respect to the circle is given by:

\[ x - 14y - 6 = 0. \]