The combine equation of the lines passing through the origin and having slopes 3 and 2 is

Options

A. `6x^(2)+5xy+y^(2)=0`
B. `6x^(2)-5xy+y^(2)=0`
C. `x^(2)+5xy+6y^(2)=0`
D. `x^(2)-5xy+6y^(2)=0`

Answer

To find the combined equation of the lines passing through the origin with slopes 3 and 2, we can follow these steps:

### Step 1: Write the equations of the lines
The equation of a line passing through the origin with slope \( m \) is given by:
\[ y = mx \]
For the slopes given in the question:
- For slope 3:
\[ L_1: y = 3x \]
- For slope 2:
\[ L_2: y = 2x \]

### Step 2: Convert the equations to standard form
We can rewrite these equations in the standard form \( Ax + By + C = 0 \):
- For \( L_1: y = 3x \):
\[ 3x - y = 0 \]
- For \( L_2: y = 2x \):
\[ 2x - y = 0 \]

### Step 3: Write the combined equation
The combined equation of the lines can be found by multiplying the two equations:
\[ (3x - y)(2x - y) = 0 \]

### Step 4: Expand the product
Now, we expand the product:
\[ 3x \cdot 2x - 3x \cdot y - 2x \cdot y + y^2 = 0 \]
This simplifies to:
\[ 6x^2 - 3xy - 2xy + y^2 = 0 \]

### Step 5: Combine like terms
Now, combine the like terms:
\[ 6x^2 - 5xy + y^2 = 0 \]

### Final Result
Thus, the combined equation of the lines is:
\[ 6x^2 - 5xy + y^2 = 0 \]

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