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

Answer

To find the chord of contact of the point (0, 5) with respect to the circle given by the equation \( x^2 + y^2 - 5x + 4y - 2 = 0 \), we will follow these steps:

### Step 1: Identify the coefficients of the circle equation
The general form of the circle's equation is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation \( x^2 + y^2 - 5x + 4y - 2 = 0 \), we can identify:
- \( 2g = -5 \) → \( g = -\frac{5}{2} \)
- \( 2f = 4 \) → \( f = 2 \)
- \( c = -2 \)

### Step 2: Write the formula for the chord of contact
The formula for 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 \( (x_1, y_1) = (0, 5) \):
\[ x(0) + y(5) + g(x + 0) + f(y + 5) + c = 0 \]

### Step 3: Substitute the values of \( g \), \( f \), \( c \), \( x_1 \), and \( y_1 \)
Substituting the values we found:
\[ 0 + 5y - \frac{5}{2}x + 2(y + 5) - 2 = 0 \]
This simplifies to:
\[ 5y - \frac{5}{2}x + 2y + 10 - 2 = 0 \]

### Step 4: Combine like terms
Combining the terms gives:
\[ 7y - \frac{5}{2}x + 8 = 0 \]

### Step 5: Clear the fraction by multiplying through by -2
To eliminate the fraction, multiply the entire equation by -2:
\[ -2(7y) + 5x - 16 = 0 \]
This simplifies to:
\[ 5x - 14y - 16 = 0 \]

### Final Answer
Thus, the equation of the chord of contact is:
\[ 5x - 14y - 16 = 0 \]