First slide

Cartesian plane

Question

$The area of the parallelogram formed by the lines y = m x, y = m x + 1, y = n x, and y = n x + 1 equals$

Moderate

A

$| m + n | / (m - n)^{2}$
B

$2 / | m + n |$
C

$1 / (| m + n |)$
D

$1 / (| m - n |)$

Solution

$y - m x i s a l i n e t h r o u g h (0, 0) a n d y - m x + 1 i s a l i n e P a r a l l e l t o y - m x h a v i n g y - i n t e r c e P t 1 . T h e v e r t i c e s a r e O (0, 0), X' A (l l (m - n), m l (m - n)) . T h e a r e a o f p a r a l l e \log r a m i s g i v e n b y$
$\begin{matrix} 2 \times Ar (Δ O A B) \\ = 2 \times \frac{1}{2} |\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 1 \\ \frac{1}{m - n} & \frac{m}{m - n} & 1 \end{array}| \\ = \frac{1}{| m - n |} \end{matrix}$

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