The straight lines represented by the equation `9x^(2)-12xy+4y^(2)=0` are

Options

A. coincedent
B. perpendicular
C. interset at `60^(@)`
D. inclined at an angle of `45^(@)`

Explanation

To solve the problem of finding the nature of the straight lines represented by the equation \(9x^2 - 12xy + 4y^2 = 0\), we can follow these steps:

### Step 1: Identify the equation
The given equation is:
\[ 9x^2 - 12xy + 4y^2 = 0 \]

### Step 2: Factor the quadratic equation
To factor the quadratic equation, we can rewrite it in a suitable form. We can recognize that this is a quadratic in \(x\) and can be factored as follows:

1. Rewrite the equation:
\[ 9x^2 - 12xy + 4y^2 = 0 \]

2. Notice that it resembles the form \(a^2 - 2ab + b^2\), which can be factored as \((a - b)^2\). Here, we can set \(a = 3x\) and \(b = 2y\):
\[ (3x - 2y)^2 = 0 \]

### Step 3: Set the factors to zero
From the factorization, we have:
\[ (3x - 2y)(3x - 2y) = 0 \]
This implies:
\[ 3x - 2y = 0 \]

### Step 4: Determine the nature of the lines
Since we have the same factor repeated, this indicates that both lines are coincident. Therefore, the lines represented by the equation are coincident lines.

### Final Answer
The straight lines represented by the equation \(9x^2 - 12xy + 4y^2 = 0\) are **coincident lines**.

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