Question

The number of possible straight lines, passing through $(2,4)$ and forming a triangle with coordinates axes, whose area is $12$ sq.units, is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (B)

$2$

Let $AB$ line is $y=mx+c$

Equation of line passing through $(2,4)$ is

$y=mx+c$ here $x=2$ and $y=4\Rightarrow c=4-2m$

$\Rightarrow y=mx+(4-2m)\Rightarrow y=m(x-2)+4$

At $(x_1,0)$

$0=m(x_1-2)+4\Rightarrow x_1=(2-\frac{4}{m})$

At $(0,y_1)$

$y_1=-2m+4$

Area of $△OAB$ is $\frac{1}{2}OA\times OB=\frac{1}{2}{x}_1\times y_1=12$

$\Rightarrow (2-\frac{4}{m})(4-2m)=24$

$\Rightarrow (2m-4)(4-2m)=24m\Rightarrow 4m^2+16m+16+24m=0$

$\Rightarrow 4m^2+8m+16=0\Rightarrow m^2+2m+4=0$

Hence $2$ value of $m$ are possible

$\therefore 2$ possible str lines