The separate equations of the lines represented by the equation `5x^(2)-9y^(2)=0` are

Options

A. `5x-3y=0 and 5x+3y=0`
B. `5x-3y=0 and 5x+9y=0`
C. `sqrt(3)x-3y=0 and sqrt(5)x+3y=0`
D. `sqrt(5)x-9y=0 and sqrt(5)x+9y=0`

Answer

To find the separate equations of the lines represented by the equation \(5x^2 - 9y^2 = 0\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 5x^2 - 9y^2 = 0 \]

### Step 2: Factor the equation
We can factor this equation using the difference of squares. We rewrite it as:
\[ ( \sqrt{5}x )^2 - (3y)^2 = 0 \]
This is in the form \(a^2 - b^2 = 0\), where \(a = \sqrt{5}x\) and \(b = 3y\).

### Step 3: Apply the difference of squares formula
Using the difference of squares formula, we have:
\[ a^2 - b^2 = (a + b)(a - b) = 0 \]
Thus, we can write:
\[ (\sqrt{5}x + 3y)(\sqrt{5}x - 3y) = 0 \]

### Step 4: Set each factor to zero
Now, we set each factor equal to zero to find the separate equations of the lines:
1. \(\sqrt{5}x + 3y = 0\)
2. \(\sqrt{5}x - 3y = 0\)

### Step 5: Simplify the equations
We can express these equations in a more standard form:
1. From \(\sqrt{5}x + 3y = 0\), we can rearrange it to:
\[ 3y = -\sqrt{5}x \quad \Rightarrow \quad y = -\frac{\sqrt{5}}{3}x \]
2. From \(\sqrt{5}x - 3y = 0\), we can rearrange it to:
\[ 3y = \sqrt{5}x \quad \Rightarrow \quad y = \frac{\sqrt{5}}{3}x \]

### Final Result
Thus, the separate equations of the lines represented by the equation \(5x^2 - 9y^2 = 0\) are:
1. \(y = -\frac{\sqrt{5}}{3}x\)
2. \(y = \frac{\sqrt{5}}{3}x\)

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