Let both root of equation `ax^2 - 2bx + 5 = 0` are `alpha` and root of equation `x^2 - 2bx - 10 = 0` are `alpha` and `beta` . Find the value of `alpha^2 + beta^ 2`

Options

A. (a)20
B. (b)25
C. (c)15
D. (d)30

Answer

To solve the problem, we need to analyze the two quadratic equations given and find the values of \(\alpha\) and \(\beta\) based on the roots of these equations.

### Step 1: Analyze the first equation
The first equation is given as:
\[ ax^2 - 2bx + 5 = 0 \]
The roots of this equation are both \(\alpha\).

Using Vieta's formulas:
- The sum of the roots is given by:
\[ \alpha + \alpha = -\frac{-2b}{a} \implies 2\alpha = \frac{2b}{a} \implies \alpha = \frac{b}{a} \]
- The product of the roots is:
\[ \alpha \cdot \alpha = \frac{5}{a} \implies \alpha^2 = \frac{5}{a} \]

### Step 2: Substitute \(\alpha\) into the product equation
Substituting \(\alpha = \frac{b}{a}\) into the product equation:
\[ \left(\frac{b}{a}\right)^2 = \frac{5}{a} \]
This simplifies to:
\[ \frac{b^2}{a^2} = \frac{5}{a} \]
Multiplying both sides by \(a^2\):
\[ b^2 = 5a \]

### Step 3: Analyze the second equation
The second equation is:
\[ x^2 - 2bx - 10 = 0 \]
The roots of this equation are \(\alpha\) and \(\beta\).

Using Vieta's formulas again:
- The sum of the roots is:
\[ \alpha + \beta = 2b \]
- The product of the roots is:
\[ \alpha \cdot \beta = -10 \]

### Step 4: Substitute \(\alpha\) into the second equation
Since \(\alpha = \frac{b}{a}\), we can substitute this into the product of the roots:
\[ \frac{b}{a} \cdot \beta = -10 \implies \beta = -\frac{10a}{b} \]

### Step 5: Find \(\alpha^2 + \beta^2\)
We know:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we have:
\[ \alpha + \beta = 2b \]
\[ \alpha \beta = -10 \]
Thus:
\[ \alpha^2 + \beta^2 = (2b)^2 - 2(-10) = 4b^2 + 20 \]

### Step 6: Substitute \(b^2\) from earlier
From our earlier calculation, we have \(b^2 = 5a\). Substituting this into the equation:
\[ \alpha^2 + \beta^2 = 4(5a) + 20 = 20a + 20 \]

### Step 7: Find the value of \(a\)
We need to find \(a\). We can use the equation derived from substituting \(\alpha\) into the second equation:
\[ \left(\frac{b}{a}\right)^2 - 2b\left(\frac{b}{a}\right) - 10 = 0 \]
Substituting \(b^2 = 5a\):
\[ \frac{5a}{a^2} - 2\frac{5a}{a} - 10 = 0 \]
This leads to:
\[ \frac{5}{a} - 10 - 10 = 0 \implies 5 - 20a = 0 \implies a = \frac{1}{4} \]

### Step 8: Substitute \(a\) back to find \(\alpha^2 + \beta^2\)
Now substituting \(a = \frac{1}{4}\) back into the equation for \(\alpha^2 + \beta^2\):
\[ \alpha^2 + \beta^2 = 20\left(\frac{1}{4}\right) + 20 = 5 + 20 = 25 \]

### Final Answer
Thus, the value of \(\alpha^2 + \beta^2\) is:
\[ \boxed{25} \]