Question
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?
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(−3, −9)
-
(−3, −8)
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(1/3, -8/3)
-
(-10/3, -7/3)
Solution
C.
(1/3, -8/3)
As the given lines x-y +1 =0 and 7x-y-5 = 0 are not parallel, therefore they represent the adjacent sides of the rhombus.
On solving x-y+1 = 0 adn 7x - y -5 = 0. we get x =1 and y =2
Thus, one of the vertex is A(1,2)
Let the coordinate of point C be (x,y)
Then, 
⇒ x+1 =- 2 and y =-4-2
⇒ x=-3 and y =-6
Hence, coordinates of C = (-3,-6)
Note that, vertices B and D will satisfy x-y +1 =0 and 7x - y-5 = 0, therefore the coordinate of vertex D is (1/3, -8/3)
