Question

A line $y=x+2$ is drawn on the co-ordinate plane. This line is rotated by ${90}^{\circ }$ clockwise about the point $(0,2).$ Another line $y=-2x+10$ is drawn such that a triangle is formed by these three lines. If the area of the triangle is $A$ sq. unit, then the value of $3A$ is

Answer 1

Given, $y=x+2...\left(1\right)$
Equation of the line when it is rotated by ${90}^{\circ }$ is
$y-2=-1(x-0)$
$\Rightarrow y=2-x...\left(2\right)$
$y=-2x+10...\left(3\right)$


Area of triangle $ABC$ is,
$A=\frac{1}{2}\left|\begin{array}{ccc}0& 2& 1\\ \frac{8}{3}& \frac{14}{3}& 1\\ 8& -6& 1\end{array}\right|$

$=\frac{1}{6}\left|\begin{array}{ccc}0& 2& 1\\ 8& 14& 3\\ 8& -6& 1\end{array}\right|$

$=\frac{1}{6}\left|\begin{array}{ccc}0& 2& 1\\ 0& 20& 2\\ 8& -6& 1\end{array}\right|$

$=\frac{1}{6}\left|8\right(4-20\left)\right|=\frac{64}{3}$

$\therefore 3A=64$