Equation of line in Straight lines

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Question

Two sides of a rectangle are 3x+ 4y + 5 =0 and4x-3y + 15 = 0. If one of its vertices is (0, 0), then the area of the rectangle is _.

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Solution

$O r i g i n i s n o t a p o i n t o n a n y o f t h e l i n e s g i v e n s o t h a t o r i g i n i s a v e r t e x w h i c h d o e s n o t l i e s o n t h e g i v e n s i d e s t h e p e r p e n d i c u l a r d i s \tan c e f r o m o r i g i n t o o n e l i n e i s l e n g t h o f t h e r e c \tan g l e a n d t h e p e r p e n d i c u a l r d i s \tan c e f r o m o r i g i n t o t h e s e c o n d l i n e i s b r e a d t h o f t h e r e c \tan g l e$ Distance of (0, 0) from the given lines is
$\begin{array}{l} a = \frac{| 0 + 0 + 5 |}{5} = 1 \\ b = \frac{| 0 - 0 + 15 |}{5} = 3 \\ Area of rectangle is a b = 3 sq. units. \end{array}$

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