If `b^(2)-4acle0` ("where" `ane0 and a,b,c,x,y in R`) satisfies the system `ax^(2)+x(b-3)+c+y=0 and ay^(2)+y(b-1)+c+3x=0`, then value of `(x)/(y)` is ___________.

Explanation

To solve the given problem step by step, we start by analyzing the equations provided in the system.

### Step 1: Write down the equations
We have two equations:
1. \( ax^2 + x(b - 3) + c + y = 0 \)
2. \( ay^2 + y(b - 1) + c + 3x = 0 \)

### Step 2: Rearranging the equations
Rearranging both equations gives us:
1. \( ax^2 + x(b - 3) + c = -y \) (Equation 1)
2. \( ay^2 + y(b - 1) + c = -3x \) (Equation 2)

### Step 3: Analyze the discriminant condition
Given that \( b^2 - 4ac < 0 \), this implies that the quadratic equations formed by both expressions will have no real roots. Therefore, both equations must be less than or equal to zero for all real \( x \) and \( y \).

### Step 4: Setting up the relationship
From the first equation, since \( ax^2 + x(b - 3) + c \leq 0 \), we can infer that:
\[ y \geq - (ax^2 + x(b - 3) + c) \]

From the second equation, since \( ay^2 + y(b - 1) + c \leq 0 \), we can infer that:
\[ 3x \geq - (ay^2 + y(b - 1) + c) \]

### Step 5: Establishing a relationship between \( x \) and \( y \)
From the rearranged equations, we can express \( y \) in terms of \( x \):
1. From Equation 1: \( y = - (ax^2 + x(b - 3) + c) \)
2. From Equation 2: \( 3x = - (ay^2 + y(b - 1) + c) \)

### Step 6: Substitute \( y \) into the second equation
Substituting the expression for \( y \) from Equation 1 into Equation 2 will give us a relationship between \( x \) and \( y \).

### Step 7: Solve the equations
By substituting and simplifying, we can find a direct relationship between \( x \) and \( y \). After some algebraic manipulation, we find:
\[ 3x = y \]

### Step 8: Finding the ratio \( \frac{x}{y} \)
From the relationship \( 3x = y \), we can express \( \frac{x}{y} \) as:
\[ \frac{x}{y} = \frac{x}{3x} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( \frac{x}{y} \) is \( \frac{1}{3} \).

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