Question

# If the straight lines $$y=4-3x; ay=x+10; 2y+bx+9=0$$ represent the three consecutive sides of a rectangle, then $$ab$$ $$=$$

A
18
B
3
C
12
D
13

Solution

## The correct option is A $$18$$Let the rectangle be $$ABCD$$The equation of the consecutive sides are given as,$$AB\rightarrow y=4-3x$$ i.e. $${ m }_{ AB }=-3$$$$BC\rightarrow ay=x+10$$ i.e. $${ m }_{ BC }=\dfrac { 1 }{ a }$$ and$$CD\rightarrow 2y+bx+9=0$$ i.e. $${ m }_{ CD }=-\dfrac { b }{ 2 }$$Now, $$AB\parallel CD$$ and $$BC\bot AB$$$$\therefore { m }_{ AB }={ m }_{ CD }$$$$\Rightarrow -3=-\dfrac { b }{ 2 }$$$$\Rightarrow b=6$$$${ m }_{ AB }\times { m }_{ BC }=-1$$$$\Rightarrow -3\times \dfrac { 1 }{ a } =-1$$$$\Rightarrow a=3$$$$\Rightarrow ab=6\times 3=18$$Mathematics

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