If the equation of a chord of the parabola `y^(2)=4ax` is `y=mx+c`, then its mid point is

Options

A. `((2a-mc)/(m^(2)),(2a)/(m))`
B. `((2a+mc)/(m^2),(2a)/(m))`
C. `((2a-mc)/(m^(2)),(-2a)/(m))`
D. `((2m-ac)/(m^(2)),(2a)/(m))`

Answer

To find the midpoint of the chord of the parabola given by the equation \( y^2 = 4ax \) and the line \( y = mx + c \), we can follow these steps:

### Step 1: Identify the points of intersection
The points of intersection of the line and the parabola can be found by substituting \( y = mx + c \) into the parabola's equation \( y^2 = 4ax \).

\[ (mx + c)^2 = 4ax \]

### Step 2: Expand and rearrange the equation
Expanding the left side gives:

\[ m^2x^2 + 2mcx + c^2 = 4ax \]

Rearranging this into standard quadratic form:

\[ m^2x^2 + (2mc - 4a)x + c^2 = 0 \]

### Step 3: Use the quadratic formula
Let the roots of this quadratic equation be \( x_1 \) and \( x_2 \). The midpoint \( x_m \) of the chord can be found using the formula for the sum of the roots:

\[ x_m = \frac{x_1 + x_2}{2} = -\frac{b}{2a} = -\frac{2mc - 4a}{2m^2} = \frac{4a - 2mc}{2m^2} = \frac{2a - mc}{m^2} \]

### Step 4: Find the corresponding y-coordinate
To find the y-coordinate of the midpoint \( y_m \), we can substitute \( x_m \) back into the equation of the line:

\[ y_m = m \left( \frac{2a - mc}{m^2} \right) + c = \frac{m(2a - mc)}{m^2} + c = \frac{2a - mc}{m} + c \]

### Step 5: Final coordinates of the midpoint
Thus, the coordinates of the midpoint \( M \) of the chord are:

\[ M\left( \frac{2a - mc}{m^2}, \frac{2a - mc}{m} + c \right) \]

### Summary of the Midpoint
The midpoint of the chord of the parabola \( y^2 = 4ax \) corresponding to the line \( y = mx + c \) is given by:

\[ M\left( \frac{2a - mc}{m^2}, \frac{2a - mc}{m} + c \right) \]