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)

The correct options are
A $x=y–1$
B $x=–(y–1)$


$A=\frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ k& k& 1\\ 2-k& k& 1\end{array}\right|$

$C_1\to C_1-C_2,C_2\to C_2-C_3$
$\Rightarrow A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 0& k-1& 1\\ 2(1-k)& k-1& 1\end{array}\right|$

$\Rightarrow A=|(k-1{)}^2|\Rightarrow (k-1{)}^2=h^2\Rightarrow k-1=\pm h$
$\Rightarrow y-1=\pm x$

$\therefore$ Locus of $P$ is
$x=y–1$ or $x=–(y–1)$