P lies on the line `y=x and Q` lies on `y= 2x`. The equation for the locus of the mid point of PQ, if `|PQ| = 4`, is `25x^2 - lambda xy + 13y^2 = 4`, then `lambda` equals

Options

A. `25x^(2)+36xy+13y^(2)=4`
B. `25x^(2)-36xy+13y^(2)=4`
C. `25x^(2)-36xy-13y^(2)=4`
D. `25x^(2)+26xy-13y^(2)=4`

Answer

To solve the problem, we need to find the value of \( \lambda \) in the equation of the locus of the midpoint of segment \( PQ \), given that \( |PQ| = 4 \). Let's break down the solution step by step.

### Step 1: Define Points P and Q
Let point \( P \) lie on the line \( y = x \). We can represent \( P \) as:
\[ P(h, h) \]
where \( h \) is the x-coordinate (and y-coordinate) of point \( P \).

Let point \( Q \) lie on the line \( y = 2x \). We can represent \( Q \) as:
\[ Q(s', 2s') \]
where \( s' \) is the x-coordinate of point \( Q \).

### Step 2: Find the Midpoint R of PQ
The midpoint \( R \) of segment \( PQ \) can be calculated using the midpoint formula:
\[ R\left(\frac{h + s'}{2}, \frac{h + 2s'}{2}\right) \]
Let \( x = \frac{h + s'}{2} \) and \( y = \frac{h + 2s'}{2} \).

### Step 3: Express h and s' in terms of x and y
From the equations for \( x \) and \( y \):
1. \( h + s' = 2x \) (1)
2. \( h + 2s' = 2y \) (2)

Now, we can manipulate these equations to express \( h \) and \( s' \) in terms of \( x \) and \( y \).

### Step 4: Solve for s'
Subtract equation (1) from equation (2):
\[ (h + 2s') - (h + s') = 2y - 2x \]
This simplifies to:
\[ s' = 2y - 2x \]

### Step 5: Solve for h
Now, substitute \( s' \) back into equation (1):
\[ h + (2y - 2x) = 2x \]
This simplifies to:
\[ h = 4x - 2y \]

### Step 6: Calculate the distance PQ
Using the distance formula, we have:
\[ |PQ|^2 = (h - s')^2 + (h - 2s')^2 \]
Substituting \( h \) and \( s' \):
\[ |PQ|^2 = \left((4x - 2y) - (2y - 2x)\right)^2 + \left((4x - 2y) - 2(2y - 2x)\right)^2 \]

### Step 7: Simplify the distance expression
Calculating the first term:
\[ (4x - 2y - 2y + 2x)^2 = (6x - 4y)^2 = 36x^2 - 48xy + 16y^2 \]
Calculating the second term:
\[ (4x - 2y - 4y + 4x)^2 = (8x - 6y)^2 = 64x^2 - 96xy + 36y^2 \]

### Step 8: Combine the terms
Now, combine both squared terms:
\[ |PQ|^2 = (36x^2 - 48xy + 16y^2) + (64x^2 - 96xy + 36y^2) \]
This simplifies to:
\[ 100x^2 - 144xy + 52y^2 \]

### Step 9: Set the distance equal to 16
Since \( |PQ| = 4 \), we have:
\[ 100x^2 - 144xy + 52y^2 = 16 \]

### Step 10: Form the equation
Multiplying through by 4 gives:
\[ 400x^2 - 576xy + 208y^2 = 64 \]
This can be rewritten as:
\[ 25x^2 - \lambda xy + 13y^2 = 4 \]

### Step 11: Compare coefficients
From the equation \( 25x^2 - \lambda xy + 13y^2 = 4 \), we can compare coefficients:
\[ \lambda = 144 \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \lambda = 36 \]