Using determinants, find the area of $\Delta PQR$ with vertices $P(3, 1), Q(9, 3)$ and $R(5, 7)$. Also, find the equation of line $PQ$ using determinants.

Area $= 16$ sq. units, Eq: $x - 3y = 0$

The correct answer is Option (3) → Area $= 16$ sq. units, Eq: $x - 3y = 0$ ##

$\text{Ar.}(\Delta PQR) = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$

$= \frac{1}{2} \begin{vmatrix} 3 & 1 & 1 \\ 9 & 3 & 1 \\ 5 & 7 & 1 \end{vmatrix}$

$= \frac{1}{2} |3(3-7) - 1(9-5) + 1(63-15)|$

$= \frac{1}{2} |-12 - 4 + 48| = 16 \text{ sq. units}$

$= \frac{32}{2} = 16 \text{ sq. units}$

Equation of the line $PQ$

$\begin{vmatrix} x & y & 1 \\ 3 & 1 & 1 \\ 9 & 3 & 1 \end{vmatrix} = 0$

$x(1 - 3) - y(3 - 9) + 1(9 - 9) = 0$

$-2x + 6y = 0$

$x - 3y = 0$