A rectangle \(A B C D\) has its side parallel to the line \(y=2 x\) and vertices \(A, B, D\) are on lines \(y=1, x=1\) and \(x=-1\) respectively. The coordinate of \(C\) can be

• A

  \((3,8)\)
• B

  \((-3,8)\)
• C

  \((-3,-1)\)
• Correct answer
  D

  None of these

Solution

Hint: Case I
line \(y=2 x\ldots(1)\) \(A D\) parallel to \(y=2 x\) \(\Rightarrow \mathrm{M}_{\mathrm{AD}}=2 ~\&~ \mathrm{M}_{\mathrm{AB}}=-\frac{1}{2}\) for \(a=3\) \(A(3,1), B(1,2), C(-3,-6), D(-1,-7)\)
\(\begin{aligned} & \text { for } a=-3 \\ & C(3,3), A(-3,1), B(1,-1), D(-1,5)\end{aligned}\)
Case II AD perpendicular to line (1) \(\Rightarrow M_{A D}=\frac{1}{2}, M_{A B}=2 \Rightarrow b=3-2 a \Rightarrow d=\frac{a+1}{2}\)
for \(a=3, C(-3,-2)\)
for \(a=-3, C(3,7)\)
\({ }^*\) In both cases if abscissa of \(C: 3\) or -3 then co-ordinates of \(C\) can be \((3,3)\) or \((3,7)\) or \((-3,-6)\) or \((-3,-2)\)

Hence, the correct answer is option: 'D'.