Joint equation of two lines, through the origin, having slopes 2 and `-2` is

Options

A. `x^(2)-4y^(2)=0`
B. `4x^(2)-y^(2)=0`
C. `x^(2)-2y^(2)=0`
D. `2x^(2)-y^(2)=0`

Answer

To find the joint equation of two lines through the origin with slopes 2 and -2, we can follow these steps:

### Step 1: Write the equations of the lines
The general equation of a line passing through the origin is given by:
\[ y = mx \]
where \( m \) is the slope of the line.

For the first line with slope \( m_1 = 2 \):
\[ y = 2x \]
We can rearrange this to:
\[ y - 2x = 0 \]

For the second line with slope \( m_2 = -2 \):
\[ y = -2x \]
We can rearrange this to:
\[ y + 2x = 0 \]

### Step 2: Combine the equations
To find the joint equation of the two lines, we multiply the two equations:
\[ (y - 2x)(y + 2x) = 0 \]

### Step 3: Expand the product
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):
\[ y^2 - (2x)^2 = 0 \]
This simplifies to:
\[ y^2 - 4x^2 = 0 \]

### Final Answer
The joint equation of the two lines through the origin with slopes 2 and -2 is:
\[ y^2 - 4x^2 = 0 \]

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