`4x^(2)-3y^(2) : 2x^(2)+5y^(2)=12 : 19`
Find `x : y=?`

Answer

To solve the problem \( 4x^2 - 3y^2 : 2x^2 + 5y^2 = 12 : 19 \) and find the ratio \( x : y \), we can follow these steps:

### Step 1: Set up the equation
We can express the given ratio in the form of a fraction:
\[ \frac{4x^2 - 3y^2}{2x^2 + 5y^2} = \frac{12}{19} \]

### Step 2: Cross-multiply
To eliminate the fraction, we cross-multiply:
\[ 19(4x^2 - 3y^2) = 12(2x^2 + 5y^2) \]

### Step 3: Distribute both sides
Now, distribute the terms on both sides:
\[ 76x^2 - 57y^2 = 24x^2 + 60y^2 \]

### Step 4: Move all terms to one side
Rearranging the equation gives us:
\[ 76x^2 - 24x^2 - 57y^2 - 60y^2 = 0 \]
This simplifies to:
\[ 52x^2 - 117y^2 = 0 \]

### Step 5: Factor the equation
We can factor out the common terms:
\[ 52x^2 = 117y^2 \]

### Step 6: Divide both sides by \( y^2 \)
Dividing both sides by \( y^2 \) gives us:
\[ \frac{52x^2}{y^2} = 117 \]

### Step 7: Solve for \( \frac{x^2}{y^2} \)
Rearranging gives:
\[ \frac{x^2}{y^2} = \frac{117}{52} \]

### Step 8: Take the square root
Taking the square root of both sides, we find:
\[ \frac{x}{y} = \sqrt{\frac{117}{52}} = \frac{\sqrt{117}}{\sqrt{52}} \]

### Step 9: Simplify the ratio
We can simplify \( \sqrt{117} \) and \( \sqrt{52} \):
\[ \sqrt{117} = 3\sqrt{13}, \quad \sqrt{52} = 2\sqrt{13} \]
Thus,
\[ \frac{x}{y} = \frac{3\sqrt{13}}{2\sqrt{13}} = \frac{3}{2} \]

### Final Answer
The ratio \( x : y \) is:
\[ x : y = 3 : 2 \]

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