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. $x=-\left(1_y\right)$

Answer 1

Answer: (A), (B)

$x=y-1$
$x=-(y-1)$

$\frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ 1& k& k\\ 1& 2-k& k\end{array}\right|=\pm h^2$
$\Rightarrow 1(k^2-k(2-k\left)\right)-1(k-k)+1(2-k-k)=\pm 2h^2$
$\Rightarrow k^2-4k+2=\pm 2h^2$
locus is ${(k-1)}^2=h^2\Rightarrow y-1=\pm x$
$x-y+1=0$ or $x+y=1$
$x=y-1$
$x=-(y-1)$