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 :

Options

A. (1,2)
B. (2,3)
C. (2,1)
D. (1,3)

Answer

To solve the problem step by step, we will analyze the given equations of the sides of the parallelogram and the diagonal, and then find the equation of the other diagonal.

### Step 1: Identify the equations of the sides of the parallelogram
The sides of the parallelogram are given by the equations:
1. \(4x + 5y = 0\) (Equation 1)
2. \(7x + 2y = 0\) (Equation 2)

### Step 2: Find the point of intersection of the two lines
To find the point of intersection, we can solve these two equations simultaneously.

From Equation 1:
\[ y = -\frac{4}{5}x \]

Substituting this value of \(y\) into Equation 2:
\[ 7x + 2\left(-\frac{4}{5}x\right) = 0 \]
\[ 7x - \frac{8}{5}x = 0 \]
Multiplying through by 5 to eliminate the fraction:
\[ 35x - 8x = 0 \implies 27x = 0 \implies x = 0 \]
Substituting \(x = 0\) back into Equation 1:
\[ 4(0) + 5y = 0 \implies y = 0 \]
Thus, the point of intersection (which is the origin) is \(O(0, 0)\).

### Step 3: Analyze the diagonal given
The equation of one of the diagonals is given as:
\[ 11x + 7y = 9 \]

### Step 4: Determine the other diagonal
Since the diagonals of a parallelogram bisect each other, we need to find the equation of the other diagonal. We know one diagonal passes through the origin \(O(0, 0)\) and the other diagonal must also pass through the midpoint of the diagonals.

### Step 5: Find the intersection of the diagonal and the sides
To find the intersection of the diagonal \(11x + 7y = 9\) with one of the sides, we can use Equation 1:
\[ 4x + 5y = 0 \implies y = -\frac{4}{5}x \]
Substituting this into the diagonal equation:
\[ 11x + 7\left(-\frac{4}{5}x\right) = 9 \]
\[ 11x - \frac{28}{5}x = 9 \]
Multiplying through by 5 to eliminate the fraction:
\[ 55x - 28x = 45 \implies 27x = 45 \implies x = \frac{45}{27} = \frac{5}{3} \]
Substituting \(x = \frac{5}{3}\) back into the equation for \(y\):
\[ y = -\frac{4}{5}\left(\frac{5}{3}\right) = -\frac{4}{3} \]
Thus, the point \(B\) is \(\left(\frac{5}{3}, -\frac{4}{3}\right)\).

### Step 6: Find the other diagonal's equation
Now, we need to find the other diagonal's equation. The other diagonal will pass through points \(O(0, 0)\) and \(B\left(\frac{5}{3}, -\frac{4}{3}\right)\).

Using the two-point formula for the equation of a line:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]
Substituting \(O(0, 0)\) as \((x_1, y_1)\) and \(B\left(\frac{5}{3}, -\frac{4}{3}\right)\) as \((x_2, y_2)\):
\[ y - 0 = \frac{-\frac{4}{3} - 0}{\frac{5}{3} - 0}(x - 0) \]
\[ y = -\frac{4}{5}x \]
This is the equation of the line through points \(O\) and \(B\).

### Step 7: Find the intersection of the other diagonal with the second side
Now, we need to find the intersection of this diagonal with the second side \(7x + 2y = 0\):
Substituting \(y = -\frac{4}{5}x\) into \(7x + 2y = 0\):
\[ 7x + 2\left(-\frac{4}{5}x\right) = 0 \]
\[ 7x - \frac{8}{5}x = 0 \]
Multiplying through by 5:
\[ 35x - 8x = 0 \implies 27x = 0 \implies x = 0 \]
Thus, the point of intersection is again the origin \(O(0, 0)\).

### Conclusion
The other diagonal passes through the point \(B\left(\frac{5}{3}, -\frac{4}{3}\right)\) and the origin. The coordinates of the other diagonal can be expressed as:
\[ \text{Other diagonal passes through } \left(-\frac{2}{3}, \frac{7}{3}\right) \]