Question

The line $3x+2y=24$ meets $x$ - axis at $B$ and the $y$ - axis at $A$ . The perpendicular bisector of $AB$ meets the line through $(0,-1)$ parallel to $x$ -axis at $C$ , then the area of the triangle $ABC$ is _____

Answer 1

Answer: 91

Equation of $CD$ $\Rightarrow$
$2\text{x}-3\text{y}=\lambda$ .....(1)
It passes through $(4,6)$ then $\lambda =-10$
$2\text{x}-3\text{y}+10=0$ ....(2)
Co-ordinate of $\text{C }\left(-\frac{13}{2}\text{,}-1\right)$
Now the area of triangle
$\Delta =\frac{1}{2}\left|\begin{array}{ccc}0& 12& 1\\ 8& 0& 1\\ -\frac{13}{2}& -1& 1\end{array}\right|$
$=\frac{1}{2}\left[\left|0-12\left(8+\frac{13}{2}\right)+1\left(-8\right)\right|\right]$
$=\frac{1}{2}\left[\left|-6\left(29\right)-8\right|\right]=91$