Straight Line and Pair of Straight Lines 3 Question 3
4. Area of the parallelogram formed by the lines $y=m x, y=m x+1, y=n x$ and $y=n x+1$ equals
(a) $\frac{|m+n|}{(m-n)^{2}}$
(b) $\frac{2}{|m+n|}$
(c) $\frac{1}{|m+n|}$
(d) $\frac{1}{|m-n|}$
(2001, 1M)
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Answer:
Correct Answer: 4. (d)
Solution:
- Let lines $O B: y=m x$
$$ \begin{aligned} C A: y & =m x+1 \\ B A: y & =n x+1 \\ \text { and } \quad O C: y & =n x \end{aligned} $$
The point of intersection $B$ of $O B$ and $A B$ has $x$ coordinate $\frac{1}{m-n}$.
Now, area of a parallelogram $O B A C$
$$ \begin{aligned} & =2 \times \text { area of } \triangle O B A \\ & =2 \times \frac{1}{2} \times O A \times D B=2 \times \frac{1}{2} \times \frac{1}{m-n} \\ & =\frac{1}{m-n}=\frac{1}{|m-n|} \end{aligned} $$
depending upon whether $m>n$ or $m<n$.
