First slide

Equation of line in Straight lines

Question

If O is the origin and $P (x_{1}, y_{1}), Q (x_{2}, y_{2})$ are two points $O P \times O Q \sin ∠ P O Q =$

Moderate

A

$x_{1} x_{2} + y_{1} y_{2}$
B

$x_{1} y_{2} + x_{2} y_{1}$
C

$|x_{1} y_{2} - x_{2} y_{1}|$
D

none of these

Solution

We, have,
Area of $ΔO P Q =$ Absolute value of $\frac{1}{2} |\begin{array}{lll} 0 0 1 \\ x_{1} y_{1} 1 \\ x_{2} y_{2} 1 \end{array}|$
$\Rightarrow$ Area of $Δ O P Q = \frac{1}{2} |x_{1} y_{2} - x_{2} y_{1}|$
Also, Area of $Δ O P Q = \frac{1}{2} O P \times O Q \times \sin ∠ P O Q$
$\begin{array}{ll} ∴ \frac{1}{2} O P \times O Q \sin ∠ P O Q = \frac{1}{2} |x_{1} y_{2} - x_{2} y_{1}| \\ \Rightarrow O P \times O Q \sin ∠ P O Q = |x_{1} y_{2} - x_{2} y_{1}| \end{array}$

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