The two adjacent sides of parallelogram are y = 0 and `y=sqrt3(x-1)`. If equation of one diagonal is `sqrt3y=(x+1)`, then equation of other diagonal is

Options

A. `sqrt(3)y = (x-1)`
B. `y = sqrt(3)(x+1)`
C. `y =- sqrt(3)(x-1)`
D. `sqrt(3)y =- (x+1)`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Identify the Given Lines
The two adjacent sides of the parallelogram are given by the equations:
1. \( y = 0 \) (which is the x-axis)
2. \( y = \sqrt{3}(x - 1) \)

The equation of one diagonal is given as:
\[ \sqrt{3}y = x + 1 \]

### Step 2: Rewrite the Diagonal Equation
We can rewrite the diagonal equation in slope-intercept form:
\[ y = \frac{1}{\sqrt{3}}x + \frac{1}{\sqrt{3}} \]

### Step 3: Determine the Slopes of the Lines
- The slope of the line \( y = 0 \) is \( 0 \).
- The slope of the line \( y = \sqrt{3}(x - 1) \) can be identified as \( \sqrt{3} \).
- The slope of the diagonal \( \sqrt{3}y = x + 1 \) is \( \frac{1}{\sqrt{3}} \).

### Step 4: Use the Properties of the Parallelogram
In a parallelogram, opposite sides are parallel, and the diagonals bisect each other. The angles formed at the intersection of the diagonals can be used to find the slope of the other diagonal.

### Step 5: Determine the Slope of the Other Diagonal
Using the properties of angles:
- The angle between the x-axis and the line \( y = \sqrt{3}(x - 1) \) is \( 60^\circ \) (since \( \tan(60^\circ) = \sqrt{3} \)).
- The angle between the diagonal \( \sqrt{3}y = x + 1 \) and the x-axis is \( 30^\circ \) (since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)).
- The other diagonal will form an angle of \( 120^\circ \) with the x-axis.

The slope of the line at \( 120^\circ \) is:
\[ \tan(120^\circ) = -\sqrt{3} \]

### Step 6: Write the Equation of the Other Diagonal
We know the slope of the other diagonal is \( -\sqrt{3} \). The equation of a line in slope-intercept form is:
\[ y = mx + c \]

Substituting the slope:
\[ y = -\sqrt{3}x + c \]

### Step 7: Find the y-intercept (c)
The other diagonal passes through the point \( (1, 0) \) (the intersection of the two adjacent sides). Substituting this point into the equation:
\[ 0 = -\sqrt{3}(1) + c \]
\[ c = \sqrt{3} \]

### Step 8: Final Equation of the Other Diagonal
Substituting \( c \) back into the equation:
\[ y = -\sqrt{3}x + \sqrt{3} \]

This can be rewritten as:
\[ y = -\sqrt{3}(x - 1) \]

### Conclusion
The equation of the other diagonal is:
\[ y = -\sqrt{3}(x - 1) \]