The equation of the chord of `y^(2) = 8x` which is bisected at (2, - 3), is

Options

A. `4x+3y=1`
B. `4x-3y=1`
C. `4x+3y=9`
D. none of these

Answer

To find the equation of the chord of the parabola \(y^2 = 8x\) that is bisected at the point \((2, -3)\), we can follow these steps:

### Step 1: Identify the parameters of the parabola
The given parabola is \(y^2 = 8x\). This can be rewritten in the standard form \(y^2 = 4ax\), where \(4a = 8\). Therefore, we find:
\[ a = 2 \]

### Step 2: Use the midpoint formula for the chord
The midpoint of the chord is given as \((x_1, y_1) = (2, -3)\). The equation of the chord of a parabola can be expressed as:
\[ y - y_1 = \frac{2a}{y_1}(x - x_1) \]
Substituting \(x_1 = 2\), \(y_1 = -3\), and \(a = 2\) into the equation, we have:
\[ y - (-3) = \frac{2 \cdot 2}{-3}(x - 2) \]
This simplifies to:
\[ y + 3 = \frac{-8}{3}(x - 2) \]

### Step 3: Simplify the equation
Now, we will simplify the equation:
\[ y + 3 = \frac{-8}{3}x + \frac{16}{3} \]
Next, we can rearrange this to isolate \(y\):
\[ y = \frac{-8}{3}x + \frac{16}{3} - 3 \]
To combine the constant terms, we convert \(3\) to a fraction:
\[ 3 = \frac{9}{3} \]
Thus, we have:
\[ y = \frac{-8}{3}x + \frac{16}{3} - \frac{9}{3} \]
This simplifies to:
\[ y = \frac{-8}{3}x + \frac{7}{3} \]

### Step 4: Rearranging to standard form
To write this in standard form \(Ax + By + C = 0\), we multiply through by 3 to eliminate the fraction:
\[ 3y = -8x + 7 \]
Rearranging gives:
\[ 8x + 3y - 7 = 0 \]

### Final Answer
The equation of the chord is:
\[ 8x + 3y - 7 = 0 \]