If `(ax^(2)+c) y+(dx^(2)+c')`=0 and `x` is `a` rational function of `y` and `ac` is `- ve` and `ac` is perfact square then

Options

A. `a^(2)+c^(2)=d^(2)+c^(2)`
B. `ad+c'=1`
C. `(a)/(c ) =(d)/(c' )`
D. `ac+dc=0`

Answer

To solve the problem step by step, we will analyze the given equation and conditions systematically.

### Given:
1. The equation: \((ax^2 + c)y + (dx^2 + c') = 0\)
2. \(x\) is a rational function of \(y\).
3. \(ac < 0\) (i.e., \(ac\) is negative).
4. \(ac\) is a perfect square.

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate terms involving \(x\):
\[ (ax^2 + c)y + (dx^2 + c') = 0 \]
This can be rewritten as:
\[ ax^2y + c y + dx^2 + c' = 0 \]
Now, we can group the terms involving \(x^2\):
\[ (ax^2y + dx^2) + (cy + c') = 0 \]
Factoring out \(x^2\):
\[ x^2(ay + d) + (cy + c') = 0 \]

### Step 2: Forming a Quadratic Equation
This is a quadratic equation in \(x\):
\[ x^2(ay + d) + (cy + c') = 0 \]
Let \(A = ay + d\) and \(C = cy + c'\). Thus, we have:
\[ Ax^2 + C = 0 \]

### Step 3: Finding the Discriminant
For \(x\) to be a rational function of \(y\), the discriminant of the quadratic equation must be a perfect square. The discriminant \(D\) is given by:
\[ D = B^2 - 4AC \]
In our case, since there is no linear term in \(x\), \(B = 0\). Therefore:
\[ D = 0^2 - 4(ay + d)(cy + c') = -4(ay + d)(cy + c') \]

### Step 4: Analyzing the Conditions
Given that \(ac < 0\) and \(ac\) is a perfect square, we need to ensure that:
1. \(-4(ay + d)(cy + c')\) is a perfect square.
2. Since \(ac < 0\), it implies one of \(a\) or \(c\) is negative.

### Step 5: Setting Up the Perfect Square Condition
For the discriminant to be a perfect square, we can set:
\[ (ay + d)(cy + c') = k^2 \quad \text{for some integer } k \]
This implies that the product of the two expressions must yield a perfect square.

### Step 6: Finding Ratios
From the conditions, we can derive:
\[ \frac{d}{c'} = \frac{a}{c} \]
This leads us to conclude that:
\[ \frac{d}{c'} = \frac{a}{c} \]

### Conclusion
The correct option based on the analysis is:
\[ \text{Option 3: } \frac{a}{c} = \frac{d}{c'} \]