If `P^2 + 4 Q^2`= 4PQ , then determine `P : Q`

Options

A. `1 :3`
B. `3 : 1`
C. `2 : 1`
D. `1 : 2`

Answer

To solve the equation \( P^2 + 4Q^2 = 4PQ \) and determine the ratio \( P : Q \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ P^2 + 4Q^2 = 4PQ \]
We can rearrange this to bring all terms to one side:
\[ P^2 + 4Q^2 - 4PQ = 0 \]

### Step 2: Recognize the quadratic form
The equation \( P^2 - 4PQ + 4Q^2 = 0 \) can be recognized as a quadratic in \( P \). We can rewrite it as:
\[ P^2 - 4PQ + 4Q^2 = 0 \]
This resembles the form \( a^2 - 2ab + b^2 = (a-b)^2 \).

### Step 3: Factor the equation
We can factor the left-hand side:
\[ (P - 2Q)^2 = 0 \]

### Step 4: Solve for \( P \)
Taking the square root of both sides gives us:
\[ P - 2Q = 0 \]
Thus, we can solve for \( P \):
\[ P = 2Q \]

### Step 5: Determine the ratio \( P : Q \)
To find the ratio \( P : Q \), we can express it as:
\[ \frac{P}{Q} = \frac{2Q}{Q} = 2 \]
This simplifies to:
\[ P : Q = 2 : 1 \]

### Final Answer
Thus, the ratio \( P : Q \) is \( 2 : 1 \).

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