Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (−1,−2), then which one of the following is a vertex of this rhombus?
Solutions · 1
Official solution
Let other two sides of rhombus are
\[x - y + \lambda = 0\]
and \[7x - y + \mu = 0\]
then O is equidistant from AB and DC and from AD and BC
\[\therefore\] \[\left| { - 1 + 2 + 1} \right| = \left| { - 1 + 2 + \lambda } \right| \Rightarrow \lambda = - 3\]
and \[\left| { - 7 + 2 - 5} \right| = \left| { - 7 + 2 + \mu } \right| \Rightarrow \mu = 15\]
\[\therefore\] Other two sides are x−y−3=0 and 7x−y+15=0
On solving the equations of sides pairwise, we get
the vertices as \[\left( {{1 \over 3},{{ - 8} \over 3}} \right),\left( {1,2} \right),\left( {{{ - 7} \over 3},{{ - 4} \over 3}} \right),\left( { - 3, - 6} \right)\]