Which of the following pair of straight lines intersect at right angles ?

Options

A. `2x^2=y(x+2y)`
B. `(x+y)^2=x(y+3x)`
C. `2y(x+y)=xy`
D. `y= overset(-)+2x`

Answer

To determine which of the given pairs of straight lines intersect at right angles, we can use the condition for perpendicular lines represented by the equation of a pair of straight lines. The general form of the equation representing a pair of straight lines through the origin is:

\[ AX^2 + 2HXY + BY^2 = 0 \]

For the lines to be perpendicular, the condition is:

\[ A + B = 0 \]

This implies that:

\[ B = -A \]

Now, let's analyze each option provided in the question.

### Step 1: Analyze the first option
**Option A:** \( 2X^2 - XY - 2Y^2 = 0 \)

Here, we can identify:
- \( A = 2 \)
- \( B = -2 \)

Now, check the condition:

\[ A + B = 2 + (-2) = 0 \]

Since this condition is satisfied, the lines represented by this equation intersect at right angles.

### Step 2: Analyze the second option
**Option B:** \( (X + Y)^2 - XY - 3X^2 = 0 \)

Expanding the left side:

\[ X^2 + Y^2 + 2XY - XY - 3X^2 = 0 \]
\[ -2X^2 + Y^2 + XY = 0 \]

Here, we identify:
- \( A = -2 \)
- \( B = 1 \)

Now, check the condition:

\[ A + B = -2 + 1 = -1 \]

Since this condition is not satisfied, the lines are not perpendicular.

### Step 3: Analyze the third option
**Option C:** \( 2Y^2 + XY = 0 \)

Rearranging gives:

\[ 0X^2 + XY + 2Y^2 = 0 \]

Here, we identify:
- \( A = 0 \)
- \( B = 2 \)

Now, check the condition:

\[ A + B = 0 + 2 = 2 \]

Since this condition is not satisfied, the lines are not perpendicular.

### Step 4: Analyze the fourth option
**Option D:** \( Y = 2X \) and \( Y = -2X \)

The slopes of the lines are:
- \( m_1 = 2 \)
- \( m_2 = -2 \)

Now, check the condition for perpendicular lines:

\[ m_1 \cdot m_2 = 2 \cdot (-2) = -4 \]

Since this condition is not equal to -1, the lines are not perpendicular.

### Conclusion
After analyzing all the options, we find that only **Option A** satisfies the condition for the lines to intersect at right angles.

### Summary of Steps:
1. Identify coefficients \( A \) and \( B \) from the equation.
2. Check the condition \( A + B = 0 \) for each option.
3. Conclude which options satisfy the condition for perpendicular lines.