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?

Options

A. (-3 , -8)
B. `((1)/(3) , - (8)/(3))`
C. `(-(10)/(3) , - (7)/(3))`
D. (-3 , -9)

Answer

B

Explanation

Let the equations of sides AB and AD be 7x - y - 5 = 0 and x - y + 1 = 0 respectively . Solving these two equations the coordinates of vertex A are (1,2)
Let the coordinates of C be `(alpha , beta)` . Since P(-1,-2) is the mid point of AC
`therefore (alpha + 1)/(2) = -1 ` and `(beta + 2)/(2) = - 2 implies alpha = -3 and beta = - 6`

So , the coordinates of C are (-3,-6)
Hence , the equations of BC and CD are x - y - 3 = 0 and 7x - y + 15 = 0 respectively . Solving equations of AD and CD ,
we find that the coordintes of D are `(-(7)/(3) , - (10)/(3))` . Since P is the mid point of BD . So , coordinate of B are `((1)/(3) , - (8)/(3))` .