Question

The area of the triangle formed by the intersection of a line parallel to $x$-axis and passing through $P(h,k)$, with the lines $y=x$ and $x+y=2$ is $h^2$. The locus of the point $P$ is

• A. x = y - 1
• B. x = - (y - 1)
• C. x = 1 + y
• D. x = - (1 + y)

Answer 1

Answer: (A), (B)

Given, ar$(△ABC)=h^2$

$\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ k& k& 1\\ 2-k& k& 1\end{array}\right|=\pm h^2$
$\Rightarrow 1(k-k)-1(k-2+k)+1\left(k^2-2k+k^2\right)=\pm 2h^2$
$\Rightarrow -(2k-2)+\left(2k^2-2k\right)=\pm 2h^2$
$\Rightarrow 2-2k+2k^2-2k=\pm 2h^2$
$\Rightarrow 2k^2-4k+2=\pm 2h^2$
$\Rightarrow k^2-2k+1=\pm h^2$
Hence, locus of a point is
$\Rightarrow (k-1{)}^2=h^2$
$y-1=\pm x$
$\Rightarrow x=y-1$ or $x=-(y-1)$