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$(\triangle ABC) = h^2$

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