Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point :
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For a parallelogram with one vertex at the origin A(0,0) and adjacent vertices B and D, the fourth vertex is C = B+D. The diagonal connecting A and C passes through the origin. The other diagonal connects B and D.
Let the two adjacent sides of the parallelogram be given by the lines:
$L_1: 4x+5y=0$
$L_2: 7x+2y=0$
These two lines intersect at one vertex of the parallelogram. Solving them simultaneously gives the vertex A:
$x=0, y=0$. So, A = (0, 0).
Let the given diagonal be $D_1: 11x+7y=9$.
This diagonal does not pass through the origin (0, 0), so it must be the diagonal connecting the other two vertices, let's say B and D. Let the parallelogram be ABCD.
Vertex C is opposite to A. The diagonals of a parallelogram bisect each other. Let the intersection point of the diagonals be P.
Let vertex B be the intersection of side AB ($L_1: 4x+5y=0$) and diagonal BD ($D_1: 11x+7y=9$).
From $L_1$, $x = -5y/4$. Substitute into $D_1$:
$11(-5y/4) + 7y = 9 \implies -55y/4 + 28y/4 = 9 \implies -27y/4 = 9 \implies y = -4/3$.
$x = -5(-4/3)/4 = 5/3$. So, B = (5/3, -4/3).
Let vertex D be the intersection of side AD ($L_2: 7x+2y=0$) and diagonal BD ($D_1: 11x+7y=9$).
From $L_2$, $y = -7x/2$. Substitute into $D_1$:
$11x + 7(-7x/2) = 9 \implies 11x - 49x/2 = 9 \implies 22x/2 - 49x/2 = 9 \implies -27x/2 = 9 \implies x = -2/3$.
$y = -7(-2/3)/2 = 7/3$. So, D = (-2/3, 7/3).
The fourth vertex C is given by the vector sum $\vec{C} = \vec{B} + \vec{D}$ (since A is the origin).
C = $(5/3 - 2/3, -4/3 + 7/3) = (3/3, 3/3) = (1, 1)$.
The other diagonal is AC. Its equation is the line passing through A(0, 0) and C(1, 1).
The equation is $y=x$.
We need to check which of the given points lies on the line $y=x$.
(A) (1, 2): $2 \neq 1$. No.
(B) (2, 2): $2 = 2$. Yes.
(C) (2, 1): $1 \neq 2$. No.
(D) (1, 3): $3 \neq 1$. No.
The other diagonal passes through the point (2, 2).
