`y=2x+c, 'c'` being variable is a chord of the parabola `y^2=4x`, meeting the parabola at A and B. Locus of a point dividing the segment AB internally in the ratio 1 : 1 is

Options

A. `y=1`
B. `x=1`
C. `y=2`
D. `x=2`

Answer

To find the locus of the point dividing the segment AB internally in the ratio 1:1, where A and B are the points of intersection of the chord \( y = 2x + c \) with the parabola \( y^2 = 4x \), we can follow these steps:

### Step 1: Find the points of intersection
We need to find the points A and B where the line \( y = 2x + c \) intersects the parabola \( y^2 = 4x \).

Substituting \( y = 2x + c \) into the parabola's equation:

\[ (2x + c)^2 = 4x \]

Expanding this gives:

\[ 4x^2 + 4cx + c^2 = 4x \]

Rearranging the equation:

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

### Step 2: Use the midpoint formula
Since the point P divides the segment AB in the ratio 1:1, it is the midpoint of A and B. The coordinates of P can be expressed as:

\[ P\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points A and B.

### Step 3: Calculate the x-coordinates of A and B
Using the quadratic formula, the x-coordinates of A and B can be found as:

\[ x = \frac{-(4c - 4) \pm \sqrt{(4c - 4)^2 - 4 \cdot 4 \cdot c^2}}{2 \cdot 4} \]

This simplifies to:

\[ x = \frac{4 - 4c \pm \sqrt{16c}}{8} = \frac{1 - c \pm \sqrt{c}}{2} \]

### Step 4: Calculate the y-coordinates of A and B
Using the line equation \( y = 2x + c \):

For \( x_1 = \frac{1 - c + \sqrt{c}}{2} \):

\[ y_1 = 2\left(\frac{1 - c + \sqrt{c}}{2}\right) + c = 1 - c + \sqrt{c} + c = 1 + \sqrt{c} \]

For \( x_2 = \frac{1 - c - \sqrt{c}}{2} \):

\[ y_2 = 2\left(\frac{1 - c - \sqrt{c}}{2}\right) + c = 1 - c - \sqrt{c} + c = 1 - \sqrt{c} \]

### Step 5: Midpoint coordinates
Now, the coordinates of point P (midpoint) are:

\[ P\left( \frac{\frac{1 - c + \sqrt{c}}{2} + \frac{1 - c - \sqrt{c}}{2}}{2}, \frac{(1 + \sqrt{c}) + (1 - \sqrt{c})}{2} \right) \]

This simplifies to:

\[ P\left( \frac{1 - c}{2}, 1 \right) \]

### Step 6: Locus of point P
As \( c \) varies, the x-coordinate \( \frac{1 - c}{2} \) changes, but the y-coordinate remains constant at 1. Thus, the locus of point P is:

\[ y = 1 \]

### Final Answer
The locus of the point dividing the segment AB internally in the ratio 1:1 is:

\[ \boxed{y = 1} \]