If P is the point (1,0) and Q lies on the parabola `y^(2)=36x`, then the locus of the mid point of PQ is :

Options

A. `y^(2)=9(2x-1)`
B. `y^(2)=9(x+2)`
C. `y^(2)=2(x-9)`
D. `x^(2)=9(y-2)`

Answer

To find the locus of the midpoint of the segment PQ, where P is the point (1, 0) and Q lies on the parabola \(y^2 = 36x\), we can follow these steps:

### Step 1: Define the Points
Let the coordinates of point P be \(P(1, 0)\) and let the coordinates of point Q on the parabola be \(Q(x_1, y_1)\).

### Step 2: Use the Parabola Equation
Since Q lies on the parabola, it satisfies the equation:
\[ y_1^2 = 36x_1 \]

### Step 3: Find the Midpoint
The midpoint M of the segment PQ can be calculated using the midpoint formula:
\[ M\left(h, k\right) = \left(\frac{x_1 + 1}{2}, \frac{y_1 + 0}{2}\right) = \left(\frac{x_1 + 1}{2}, \frac{y_1}{2}\right) \]
Thus, we have:
\[ h = \frac{x_1 + 1}{2} \quad \text{and} \quad k = \frac{y_1}{2} \]

### Step 4: Express \(x_1\) and \(y_1\) in terms of \(h\) and \(k\)
From the equations for h and k, we can express \(x_1\) and \(y_1\) as:
\[ x_1 = 2h - 1 \quad \text{and} \quad y_1 = 2k \]

### Step 5: Substitute into the Parabola Equation
Now substitute \(x_1\) and \(y_1\) into the parabola equation:
\[ (2k)^2 = 36(2h - 1) \]
This simplifies to:
\[ 4k^2 = 36(2h - 1) \]

### Step 6: Simplify the Equation
Dividing both sides by 4 gives:
\[ k^2 = 9(2h - 1) \]

### Step 7: Replace \(h\) and \(k\) with \(x\) and \(y\)
Since \(h\) corresponds to \(x\) and \(k\) corresponds to \(y\), we replace them:
\[ y^2 = 9(2x - 1) \]

### Final Result
The locus of the midpoint M of segment PQ is given by:
\[ y^2 = 9(2x - 1) \]