Question

The equations of the sides $AB , BC$ and $CA$ of a triangle $ABC$ are $2 x + y =0, x + py =39$ and $x - y =3$ respectively and $P (2,3)$ is its circumcentre. Then which of the following is NOT true :

• A. $( AC )^2=9 p$
• B. $( AC )^2+ p ^2=136$
• C. $32<\operatorname{area}(\triangle ABC )<36$
• D. $34<\operatorname{area}(\triangle ABC )<38$

Answer 1

Answer: (D)

Perpendicular bisector of $A B$
$x+y=5$
Take image of A
$ \frac{ x -1}{1}=\frac{ y +2}{1}=\frac{-2(-6)}{2}=6$
$(7,4) $
$ 7+4 p =39$
$p =8$
solving $ x +8 y =39 \text { and } y =-2 x$
$ x =\frac{-39}{15} y =\frac{78}{15} $
$ AC ^2=72=9 p$
$AC ^2+ p ^2=72+64=136$
$ \Delta ABC =\frac{1}{2} \begin{vmatrix} 1 & -2 & 1 \\7 & 4 & 1 \\\frac{-39}{15} & \frac{78}{15} & 1\end{vmatrix}$
$ =\frac{1}{2}\left[4-\frac{78}{15}+2\left(7+\frac{39}{15}\right)+7\left(\frac{78}{15}\right)+\frac{4 \times 39}{15}\right] $
$=\frac{1}{2}\left[18+18 \times \frac{13}{5}\right] $
$ =9\left[\frac{18}{5}\right]=\frac{162}{5}=32.4$