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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
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B
3
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C
12
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D
13
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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 $$ BC\bot AB$$
$$ \therefore  { m }_{ AB }={ m }_{ CD }$$

$$ \Rightarrow -3=-\dfrac { b }{ 2 } $$
$$\Rightarrow b=6$$

$${ m }_{ AB }\times  { m }_{ BC }=-1$$
$$\Rightarrow -3\times \dfrac { 1 }{ a } =-1$$

$$\Rightarrow a=3$$
$$ \Rightarrow  ab=6\times 3=18$$

99746_94711_ans.png

Mathematics

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