Let Q and R be two points on the line x + 1 2 = y + 2 3 = z - 1 2 at a distance 26 from the point P 4 , 2 ,…

Given,

$L:\frac{x+1}{2}=\frac{y+2}{3}=\frac{2-1}{2}$

Plotting the diagram of given value's in question we have,

Let $T$ be any point on line whose coordinates are  $T\left(2t-1,3t-2,2t+1\right)$

Now $PT\perp QR$ by diagram,

So, $2\left(2t-5\right)+3\left(3t-4\right)+2\left(2t-6\right)=0$

$\Rightarrow 17t=34$

$\Rightarrow t=2$, so $T\left(3,4,5\right)$

Now the value of $\left|PT\right|=\sqrt{1+4+4}=3$

And $\left|QT\right|=\sqrt{26-9}=\sqrt{17}$ {by using pythagorus theorem}

$\therefore$ Area of $∆PQR=\frac{1}{2}\times 2\sqrt{17}\times 3=3\sqrt{17}$

$\therefore$ Square of $ar\left(∆PQR\right)=153.$