If one root of the equation `x^(2) + ax + b =0` is 1/3 times the other. Then, the correct relation between a and b

Options

A. `3a^(2) = 16b`
B. `16a^(2) = 3b`
C. `3a = 16b^(2)`
D. `16a = 3b^(2)`

Answer

To solve the problem, we need to establish a relationship between the coefficients \( a \) and \( b \) of the quadratic equation \( x^2 + ax + b = 0 \) given that one root is \( \frac{1}{3} \) times the other.

Let's denote the two roots of the equation as follows:

1. Let the first root be \( \alpha \).
2. The second root will then be \( \frac{\alpha}{3} \).

### Step 1: Sum of the Roots
According to Vieta's formulas, the sum of the roots of the quadratic equation \( x^2 + ax + b = 0 \) is given by:

\[ \text{Sum of roots} = -\frac{a}{1} = -a \]

Now, substituting our roots into this formula:

\[ \alpha + \frac{\alpha}{3} = -a \]

### Step 2: Simplifying the Sum of Roots
To simplify the left side:

\[ \alpha + \frac{\alpha}{3} = \frac{3\alpha}{3} + \frac{\alpha}{3} = \frac{4\alpha}{3} \]

So, we have:

\[ \frac{4\alpha}{3} = -a \]

From this, we can express \( \alpha \):

\[ \alpha = -\frac{3a}{4} \]

### Step 3: Product of the Roots
Next, we use Vieta's formulas for the product of the roots, which states:

\[ \text{Product of roots} = \frac{b}{1} = b \]

Now substituting our roots into this formula:

\[ \alpha \cdot \frac{\alpha}{3} = b \]

### Step 4: Substituting the Value of \( \alpha \)
Substituting \( \alpha = -\frac{3a}{4} \) into the product equation:

\[ \left(-\frac{3a}{4}\right) \cdot \left(-\frac{3a}{4 \cdot 3}\right) = b \]

This simplifies to:

\[ \left(-\frac{3a}{4}\right) \cdot \left(-\frac{a}{4}\right) = b \]

### Step 5: Simplifying the Product
Calculating the left side:

\[ \frac{3a^2}{16} = b \]

### Final Relation
Thus, we arrive at the relationship between \( a \) and \( b \):

\[ 3a^2 = 16b \]

### Conclusion
The correct relation between \( a \) and \( b \) is:

\[ 3a^2 = 16b \]