`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 of the chord \( y = 2x + c \) of the parabola \( y^2 = 4x \) internally in the ratio 1:1, we will follow these steps:

### Step 1: Find the points of intersection of the chord and the parabola

The equation of the parabola is given by:
\[ y^2 = 4x \]
The equation of the chord is:
\[ y = 2x + c \]
To find the points of intersection (A and B), substitute \( y \) from the chord equation into the parabola equation:
\[ (2x + c)^2 = 4x \]
Expanding this gives:
\[ 4x^2 + 4cx + c^2 = 4x \]
Rearranging it leads to:
\[ 4x^2 + (4c - 4)x + c^2 = 0 \]

### Step 2: Use the quadratic formula to find the x-coordinates of A and B

The roots of the quadratic equation \( 4x^2 + (4c - 4)x + c^2 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = 4c - 4 \), and \( c = c^2 \). Thus, we have:
\[ x = \frac{-(4c - 4) \pm \sqrt{(4c - 4)^2 - 4 \cdot 4 \cdot c^2}}{2 \cdot 4} \]
Simplifying this gives:
\[ x = \frac{4 - 4c \pm \sqrt{16(c - 1)^2}}{8} \]
\[ x = \frac{4 - 4c \pm 4|c - 1|}{8} \]
This results in two cases for \( x \).

### Step 3: Find the mid-point of segment AB

Let the x-coordinates of points A and B be \( x_1 \) and \( x_2 \). The midpoint \( M \) of segment AB is given by:
\[ M_x = \frac{x_1 + x_2}{2} \]
Since the sum of the roots \( x_1 + x_2 \) is given by \( -\frac{b}{a} = -\frac{4c - 4}{4} = 1 - c \), we have:
\[ M_x = \frac{(1 - c)}{2} \]

### Step 4: Find the y-coordinates of points A and B

To find the y-coordinates, substitute \( M_x \) back into the chord equation:
\[ M_y = 2M_x + c = 2\left(\frac{1 - c}{2}\right) + c = 1 - c + c = 1 \]

### Step 5: Write the coordinates of the midpoint M

Thus, the coordinates of the midpoint \( M \) are:
\[ M\left(\frac{1 - c}{2}, 1\right) \]

### Step 6: Find the locus of point M as \( c \) varies

To find the locus, we eliminate \( c \). From the x-coordinate:
\[ x = \frac{1 - c}{2} \implies c = 1 - 2x \]
Substituting this into the y-coordinate:
\[ y = 1 \]

### Final Result

The locus of the point dividing the segment AB internally in the ratio 1:1 is:
\[ y = 1 \]