The equation `(ay-bx)^(2)+4xy=0` has rational solutions `x`, `y` for

Options

A. `a=(1)/(2)`,`b=2`
B. `a=4`,`b=(1)/(8)`
C. `a=1`,`b=(3)/(4)`
D. `a=2`,`b=1`

Explanation

To solve the equation \((ay - bx)^2 + 4xy = 0\) for rational solutions \(x\) and \(y\), we will analyze the equation step by step.

### Step 1: Analyze the given equation
The equation is \((ay - bx)^2 + 4xy = 0\). Since a square is always non-negative, for the sum to be zero, both terms must be zero. Therefore, we can set:
\[ (ay - bx)^2 = 0 \quad \text{and} \quad 4xy = 0 \]

### Step 2: Solve the first equation
From \((ay - bx)^2 = 0\), we have:
\[ ay - bx = 0 \implies ay = bx \implies \frac{y}{x} = \frac{b}{a} \]
This gives us a relationship between \(y\) and \(x\).

### Step 3: Solve the second equation
From \(4xy = 0\), we have:
\[ xy = 0 \]
This implies either \(x = 0\) or \(y = 0\).

### Step 4: Consider the cases
1. **Case 1**: If \(x = 0\), then from \(ay = bx\), we have \(ay = 0\). Thus, \(y\) must also be \(0\).
2. **Case 2**: If \(y = 0\), then from \(ay = bx\), we have \(0 = bx\). Thus, \(x\) must also be \(0\).

In both cases, we find that \(x = 0\) and \(y = 0\) is a trivial solution.

### Step 5: Check for non-trivial rational solutions
To find non-trivial rational solutions, we substitute \(y = kx\) (where \(k\) is a rational number) into the equation:
\[ (akx - bx)^2 + 4x(kx) = 0 \]
This simplifies to:
\[ x^2(a^2k^2 - 2abk + 4k) = 0 \]
For non-trivial solutions, we need:
\[ a^2k^2 - 2abk + 4k = 0 \]
This is a quadratic in \(k\).

### Step 6: Find the discriminant
The discriminant \(D\) of the quadratic \(a^2k^2 - (2ab - 4)k = 0\) must be a perfect square for \(k\) to be rational:
\[ D = (2ab - 4)^2 - 4a^2 \cdot 0 = (2ab - 4)^2 \]
Setting this equal to a perfect square gives us conditions on \(a\) and \(b\).

### Step 7: Rational solutions
We need to find values of \(a\) and \(b\) such that the discriminant is a perfect square.

### Conclusion
After analyzing the options:
1. \(a = \frac{1}{2}, b = 2\)
2. \(a = 4, b = \frac{1}{8}\)
3. \(a = 1, b = \frac{3}{4}\)
4. \(a = 2, b = 1\)

We can conclude that options 1 and 3 yield rational solutions based on the discriminant condition.