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. $\frac{1}{2}$
• D. $-\frac{1}{3}$

Answer 1

The correct option is A $18$

Let the rectangle be $ABCD$

The equation of the consecutive sides are given as,

$AB\to y=4-3x$ i.e. $m_{AB}=-3$

$BC\to ay=x+10$ i.e. $m_{BC}=\frac{1}{a}$ and

$CD\to 2y+bx+9=0$ i.e. $m_{CD}=-\frac{b}{2}$

Now, $AB\parallel CD$ and $BC\perp AB$

$\therefore m_{AB}=m_{CD}$

$\Rightarrow -3=-\frac{b}{2}$

$\Rightarrow b=6$

$m_{AB}\times m_{BC}=-1$

$\Rightarrow -3\times \frac{1}{a}=-1$

$\Rightarrow a=3$

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