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Question

Mathematics Question on Straight lines

Two sides of a rhombus are along the lines, xy+1=0x - y + 1 = 0 and 7xy5=07x - y - 5 = 0. If its diagonals intersect at (1,2)(-1, -2), then which one of the following is a vertex of this rhombus?

A

(-3 , -9)

B

(-3 , -8)

C

(13,83)\left( \frac{1}{3} , - \frac{8}{3} \right)

D

(103,73)\left( - \frac{10}{3} , - \frac{7}{3} \right)

Answer

(13,83)\left( \frac{1}{3} , - \frac{8}{3} \right)

Explanation

Solution

Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0

Coordinates of A(1,2)A \equiv (1, 2)
\therefore Slope of AE=2AE = 2
\Rightarrow Slope of BD=12BD = - \frac{1}{2} \Rightarrow
E of BDBD is y+2x+1=12\frac{y + 2 }{ x +1 } = - \frac{1}{2}
x+2y+5=0\Rightarrow x + 2y + 5 = 0
\therefore Co-ordinates of D=(13,83)D = \left( \frac{1}{3} , \frac{-8}{3} \right)

So, the correct option is (C): (13,83)\left( \frac{1}{3} , - \frac{8}{3} \right)