If the roots of the equation `ax^(2) + bx + c= 0` is `1/k` times the roots of `px^(2) + qx + r = 0`, then which of the following is true ?

Options

A. `a= pk`
B. `a/b = p/q`
C. `aq = pbk`
D. `ab = pqk^(2)`

Explanation

To solve the problem step-by-step, let's analyze the given information and derive the necessary relationships.

### Step 1: Understand the given equations
We have two quadratic equations:
1. \( ax^2 + bx + c = 0 \)
2. \( px^2 + qx + r = 0 \)

We are given that the roots of the first equation are \( \frac{1}{k} \) times the roots of the second equation.

### Step 2: Express the roots of the second equation
Let the roots of the second equation \( px^2 + qx + r = 0 \) be \( \alpha \) and \( \beta \). According to the problem, the roots of the first equation will be \( \frac{\alpha}{k} \) and \( \frac{\beta}{k} \).

### Step 3: Use the relationship of roots and coefficients
From Vieta's formulas, we know:
- For the second equation:
- Sum of roots \( \alpha + \beta = -\frac{q}{p} \)
- Product of roots \( \alpha \beta = \frac{r}{p} \)

- For the first equation:
- Sum of roots \( \frac{\alpha}{k} + \frac{\beta}{k} = \frac{1}{k}(\alpha + \beta) = -\frac{q}{kp} \)
- Product of roots \( \frac{\alpha}{k} \cdot \frac{\beta}{k} = \frac{\alpha \beta}{k^2} = \frac{r}{pk^2} \)

### Step 4: Relate the coefficients
Using Vieta's formulas for the first equation, we can express the coefficients in terms of \( a, b, c \):
- The sum of the roots gives us:
\[ -\frac{b}{a} = -\frac{q}{kp} \implies b = \frac{aq}{k} \]

- The product of the roots gives us:
\[ \frac{c}{a} = \frac{r}{pk^2} \implies c = \frac{ar}{pk^2} \]

### Step 5: Establish relationships between coefficients
From the equations derived:
1. \( a = pk^2 \)
2. \( b = \frac{aq}{k} \)
3. \( c = \frac{ar}{pk^2} \)

### Step 6: Verify the options
Now we can check the relationships to determine which statement is true:
- **Option A**: \( a = pk \) (This is false, as we have \( a = pk^2 \))
- **Option B**: \( \frac{a}{b} = \frac{pk^2}{\frac{aq}{k}} = \frac{pk^3}{a} \) (This is also false)
- **Option C**: \( aq = pbk \) (This can be derived from the relationships we have established)

### Conclusion
The correct statement that holds true based on our derivation is:
**Option C: \( aq = pbk \)**