The quadratic equation whose roots are reciprocal of the roots of the equation `ax^(2) + bx+c=0` is :

Options

A. `cx^(2)+bx+a=0`
B. `bx^(2)+cx+a=0`
C. `cx^(2)+ax+b=0`
D. `bx^(2)+ax+c=0`

Answer

To find the quadratic equation whose roots are the reciprocals of the roots of the given quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the roots of the original equation
Let the roots of the equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 3: Determine the new roots
The new roots we are interested in are the reciprocals of the original roots, which are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \).

### Step 4: Calculate the sum of the new roots
The sum of the new roots is:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} = -\frac{b}{c} \]

### Step 5: Calculate the product of the new roots
The product of the new roots is:
\[ \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha \beta} = \frac{1}{\frac{c}{a}} = \frac{a}{c} \]

### Step 6: Form the new quadratic equation
Using the sum and product of the new roots, we can form the new quadratic equation:
\[ x^2 - \left(\text{sum of roots}\right)x + \left(\text{product of roots}\right) = 0 \]
Substituting the values we found:
\[ x^2 - \left(-\frac{b}{c}\right)x + \frac{a}{c} = 0 \]
This simplifies to:
\[ x^2 + \frac{b}{c}x + \frac{a}{c} = 0 \]

### Step 7: Clear the fraction by multiplying through by \( c \)
Multiplying the entire equation by \( c \) to eliminate the fraction gives:
\[ cx^2 + bx + a = 0 \]

### Final Result
The quadratic equation whose roots are the reciprocals of the roots of the equation \( ax^2 + bx + c = 0 \) is:
\[ cx^2 + bx + a = 0 \]

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