The equation of the chord of contact of tangents from (2, 5) to the parabola `y^(2)=8x,` is

Options

A. `4x+5y+8=0`
B. `4x-5y+8=0`
C. `4x-5y-9=0`
D. `4x+5y-8=0`

Explanation

To find the equation of the chord of contact of tangents from the point (2, 5) to the parabola \( y^2 = 8x \), we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 8x \). From this equation, we can identify that:
- The value of \( 4a = 8 \), which gives \( a = 2 \).

**Hint:** The parameter \( a \) in the parabola \( y^2 = 4ax \) represents the distance from the vertex to the focus.

### Step 2: Use the formula for the chord of contact
The formula for the chord of contact of tangents from a point \( (x_1, y_1) \) to the parabola \( y^2 = 4ax \) is given by:
\[ yy_1 = 2a(x + x_1) \]
In our case, \( (x_1, y_1) = (2, 5) \) and \( a = 2 \).

**Hint:** Remember to substitute the coordinates of the point and the value of \( a \) into the formula.

### Step 3: Substitute the values into the formula
Substituting \( x_1 = 2 \), \( y_1 = 5 \), and \( a = 2 \) into the chord of contact formula:
\[ y \cdot 5 = 2 \cdot 2 (x + 2) \]
This simplifies to:
\[ 5y = 8(x + 2) \]

**Hint:** Make sure to simplify the equation correctly after substituting the values.

### Step 4: Expand and rearrange the equation
Expanding the equation gives:
\[ 5y = 8x + 16 \]
Now, rearranging it to standard form:
\[ 8x - 5y + 16 = 0 \]

**Hint:** Rearranging the equation helps in identifying the standard form of the line.

### Final Answer
The equation of the chord of contact of tangents from the point (2, 5) to the parabola \( y^2 = 8x \) is:
\[ 8x - 5y + 16 = 0 \]