Question

If $\ell ,m,n$ are three consecutive odd integers then the family of lines $\ell x-my+n=0$ passes through a fixed point $M(a,b)$
Number of straight lines passing through $M(a,b)$ and making an area of $2$ square unit with coordinate axes in the first quadrant, is equal to

• A. 1
• B. 0
• C. 2
• D. 4

Answer 1

Answer: (B)

Let $\ell =2r+1,m=2r+3,n=2r+5$
$\therefore 2r(x-y+1)+(x-3y+5)=0$
$\Rightarrow$ line passes through fixed point $(1,2)$
So, $a=1,b=2$
$\Rightarrow M(1,2)$
(i) We have $\frac{x}{2h}+\frac{y}{2k}=1$ (intercept form)
$\Rightarrow \frac{1}{2x}+\frac{2}{2y}=1$
$\Rightarrow y+2x=2xy$
$\Rightarrow 2x+y-2xy=0$
(ii) Let slope of line be ' $m$ ' as $m<0$.
$\therefore$ Equation of line $(y-2)=m(x-1)$ (given)
(ii) Let slope of line be ' $m$ ' as $m<0$.
$\therefore$ Equation of line $(y-2)=m(x-1)$ (given)

Now, area of $(\Delta OAB=2)($ given $)$
$\Rightarrow \frac{1}{2}(2-m)\frac{(m-2)}{m}=2$
$\Rightarrow (m-2{)}^2=-4m$
$\Rightarrow m^2-4m+4=-4m$
$\Rightarrow m^2+4=0$
So, no real value of $m$ exist.