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  4. A rectangle ABCD has its side AB parallel to line y = x and vertices A, B and D lie on y = 1, x = 2 and x = - 2, respectively. Locus of vertex ‘C’ is

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Q. A rectangle $ABCD$ has its side $AB$ parallel to line $y = x$ and vertices $A$, $B$ and $D$ lie on $y = 1$, $x = 2$ and $x = - 2$, respectively. Locus of vertex ‘$C$’ is

UPSEEUPSEE 2013

Solution:

Let the equation of side $B C$ be $y=x+a$. $\Rightarrow A=(1-a, 1), B=(2,2+a)$
Equation of side $A D$ is $y-1=-[(x-(1-a)]$
$\Rightarrow D \equiv(-2,4-a)$
Let $C \equiv(h, k) \Rightarrow h+1-a=2-2$
$\Rightarrow h=a-1$ and $k+1=2+a+4-a \Rightarrow k=5$
Thus, the locus of $C$ is $y=5$