Find the equation of the chord of contact of the point (5, 1) to the hyperbola, `x^(2) - 4y^(2) = 16`. Also find the mid-point of this chord.

Answer

To find the equation of the chord of contact from the point \( (5, 1) \) to the hyperbola given by the equation \( x^2 - 4y^2 = 16 \), we will follow these steps:

### Step 1: Write the standard form of the hyperbola
The given hyperbola is \( x^2 - 4y^2 = 16 \). We can rewrite this in standard form:

\[ \frac{x^2}{16} - \frac{y^2}{4} = 1 \]

This indicates that the hyperbola is centered at the origin with \( a^2 = 16 \) and \( b^2 = 4 \).

### Step 2: Use the chord of contact formula
The chord of contact from a point \( (x_1, y_1) \) to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is given by the equation:

\[ \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \]

Here, \( (x_1, y_1) = (5, 1) \), \( a^2 = 16 \), and \( b^2 = 4 \).

### Step 3: Substitute the values into the chord of contact formula
Substituting the values into the formula, we get:

\[ \frac{5x}{16} - \frac{1y}{4} = 1 \]

### Step 4: Simplify the equation
To eliminate the fractions, we can multiply the entire equation by 16:

\[ 5x - 4y = 16 \]

This is the equation of the chord of contact.

### Step 5: Find the mid-point of the chord
To find the mid-point of the chord, we can use the property that the mid-point of the chord of contact lies on the line joining the point \( (5, 1) \) and the center of the hyperbola \( (0, 0) \).

The mid-point \( M(x_m, y_m) \) can be calculated as follows:

\[ x_m = \frac{x_1 + 0}{2} = \frac{5 + 0}{2} = \frac{5}{2} \]
\[ y_m = \frac{y_1 + 0}{2} = \frac{1 + 0}{2} = \frac{1}{2} \]

Thus, the mid-point of the chord is \( M\left(\frac{5}{2}, \frac{1}{2}\right) \).

### Final Answers:
1. The equation of the chord of contact is \( 5x - 4y = 16 \).
2. The mid-point of the chord is \( M\left(\frac{5}{2}, \frac{1}{2}\right) \).