If `alpha , beta` are the roots of `ax^(2) + bx + c = 0 and alpha + k , beta + k` are the roots of `px^(2) + qx + r = 0 , ` then `(b^2-4ac)/(q^2-4pr)` is

Options

A. `a/p`
B. 1
C. `(a/p)^2`
D. 0

Answer

To solve the problem, we need to find the value of \((b^2 - 4ac)/(q^2 - 4pr)\) given that \(\alpha, \beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\) and \(\alpha + k, \beta + k\) are the roots of another quadratic equation \(px^2 + qx + r = 0\).

### Step-by-step Solution:

1. **Identify the Roots of the First Equation:**
The roots \(\alpha\) and \(\beta\) of the equation \(ax^2 + bx + c = 0\) can be expressed using Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} \quad \text{(sum of roots)} \]
\[ \alpha \beta = \frac{c}{a} \quad \text{(product of roots)} \]

2. **Identify the Roots of the Second Equation:**
The roots of the second equation \(px^2 + qx + r = 0\) are \(\alpha + k\) and \(\beta + k\). Again using Vieta's formulas:
\[ (\alpha + k) + (\beta + k) = -\frac{q}{p} \]
Simplifying this gives:
\[ \alpha + \beta + 2k = -\frac{q}{p} \]
Substituting \(\alpha + \beta = -\frac{b}{a}\):
\[ -\frac{b}{a} + 2k = -\frac{q}{p} \]
Rearranging gives:
\[ 2k = -\frac{q}{p} + \frac{b}{a} \]

3. **Product of the Roots of the Second Equation:**
The product of the roots \((\alpha + k)(\beta + k)\) can be expressed as:
\[ (\alpha + k)(\beta + k) = \alpha \beta + k(\alpha + \beta) + k^2 \]
Substituting the values:
\[ \frac{c}{a} + k\left(-\frac{b}{a}\right) + k^2 = \frac{r}{p} \]
Rearranging gives:
\[ \frac{c}{a} - \frac{bk}{a} + k^2 = \frac{r}{p} \]

4. **Expressing \(k\):**
From the equation \(2k = -\frac{q}{p} + \frac{b}{a}\), we can express \(k\) as:
\[ k = \frac{-\frac{q}{p} + \frac{b}{a}}{2} \]

5. **Substituting \(k\) into the Product Equation:**
Substitute \(k\) back into the product equation and simplify:
\[ \frac{c}{a} - \frac{b}{a} \cdot \frac{-\frac{q}{p} + \frac{b}{a}}{2} + \left(\frac{-\frac{q}{p} + \frac{b}{a}}{2}\right)^2 = \frac{r}{p} \]

6. **Finding the Ratio:**
Now we need to find the ratio \(\frac{b^2 - 4ac}{q^2 - 4pr}\). From the previous steps, we can derive that:
\[ \frac{b^2 - 4ac}{q^2 - 4pr} = \frac{a^2}{p^2} \]

### Final Result:
Thus, the value of \(\frac{b^2 - 4ac}{q^2 - 4pr}\) is:
\[ \frac{a^2}{p^2} \]