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Step-by-Step Solution
Step 1: Identify the given points and the line equation
We have two fixed points:
A(1, −1) and B(0, 2).
A point P with coordinates (x', y') lies on the line
$3x + y - 4\lambda = 0$.
Step 2: Express the condition for point P on the given line
Since P lies on $3x + y - 4\lambda = 0$, if P has coordinates $(x', y')$, they must satisfy:
$3x' + y' - 4\lambda = 0 \quad \Longrightarrow \quad y' = 4\lambda - 3x'$
Step 3: Write the formula for the area of the triangle $PAB$
The area of triangle formed by points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be found using the determinant formula:
$ \text{Area} = \frac{1}{2} \left|\begin{matrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{matrix}\right| $
Here, the points are $A(1, -1)$, $B(0, 2)$, and $P(x', y')$. Given the area of $\triangle PAB$ is 5, we set up the equation:
$ \frac{1}{2} \left|\begin{matrix}
1 & -1 & 1 \\
0 & 2 & 1 \\
x' & y' & 1
\end{matrix}\right| = 5 $
Step 4: Simplify the determinant
Compute the determinant:
$\displaystyle
\left|\begin{matrix}
1 & -1 & 1 \\
0 & 2 & 1 \\
x' & y' & 1
\end{matrix}\right|
= 1 \cdot \left|\begin{matrix}2 & 1 \\ y' & 1\end{matrix}\right|
-(-1) \cdot \left|\begin{matrix}0 & 1 \\ x' & 1\end{matrix}\right|
+ 1 \cdot \left|\begin{matrix}0 & 2 \\ x' & y'\end{matrix}\right|
$
$= 1 \cdot (2 \cdot 1 - 1 \cdot y') + 1 \cdot (0 \cdot 1 - 1 \cdot x') + 1 \cdot (0 \cdot y' - 2 \cdot x')$
$= (2 - y') - x' - 2x'$
$= 2 - y' - 3x'$
Step 5: Use the area condition
We have:
$\frac{1}{2} |2 - y' - 3x'| = 5 \quad \Longrightarrow \quad |2 - y' - 3x'| = 10$
Step 6: Substitute $y' = 4\lambda - 3x'$ into the area equation
From $3x' + y' - 4\lambda = 0$, we have $y' = 4\lambda - 3x'$. Substitute this into $2 - y' - 3x'$:
$2 - (4\lambda - 3x') - 3x' = 2 - 4\lambda + 3x' - 3x' = 2 - 4\lambda$
Thus the area condition becomes:
$|\,2 - 4\lambda\,| = 10$
Step 7: Solve for $\lambda$
We solve the absolute value equation:
$2 - 4\lambda = 10 \quad \text{or} \quad 2 - 4\lambda = -10$
Case 1: If $2 - 4\lambda = 10$,
$-4\lambda = 8 \quad \Longrightarrow \quad \lambda = -2
Case 2: If $2 - 4\lambda = -10$,
$-4\lambda = -12 \quad \Longrightarrow \quad \lambda = 3
Step 8: Conclude the required value of $\lambda$
The possible values of $\lambda$ are $-2$ and $3$. According to the given options, the required value is $\lambda = 3$.
