The joint equation of the line `x-y=0 and x+y=0` is

Options

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

Answer

To find the joint equation of the lines given by the equations \(x - y = 0\) and \(x + y = 0\), we will follow these steps:

### Step 1: Write down the equations of the lines
The equations of the lines are:
1. \(L_1: x - y = 0\)
2. \(L_2: x + y = 0\)

### Step 2: Rearrange the equations
We can rearrange both equations to express them in a standard form:
1. \(L_1: x - y = 0\) can be written as \(x - y + 0 = 0\)
2. \(L_2: x + y = 0\) can be written as \(x + y + 0 = 0\)

### Step 3: Multiply the equations
To find the joint equation, we multiply the two equations:
\[ (x - y)(x + y) = 0 \]

### Step 4: Expand the product
Now, we will expand the product:
\[ x^2 - y^2 = 0 \]

### Step 5: Write the final joint equation
The final joint equation of the lines \(x - y = 0\) and \(x + y = 0\) is:
\[ x^2 - y^2 = 0 \]

### Summary
Thus, the joint equation of the lines \(x - y = 0\) and \(x + y = 0\) is \(x^2 - y^2 = 0\).

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