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. 20
B. 25
C. 15
D. 30

Answer

To solve the problem, we need to find the value of \( \alpha^2 + \beta^2 \) given the equations:

1. \( ax^2 - 2bx + 5 = 0 \) with roots \( \alpha \) and \( \alpha \) (both roots are the same).
2. \( x^2 - 2bx - 10 = 0 \) with roots \( \alpha \) and \( \beta \).

### Step 1: Analyze the first equation

From the first equation \( ax^2 - 2bx + 5 = 0 \):

- The sum of the roots \( \alpha + \alpha = 2\alpha = \frac{2b}{a} \) implies \( \alpha = \frac{b}{a} \).
- The product of the roots \( \alpha \cdot \alpha = \alpha^2 = \frac{5}{a} \).

### Step 2: Express \( \alpha^2 \) in terms of \( b \) and \( a \)

From \( \alpha = \frac{b}{a} \):

\[ \alpha^2 = \left(\frac{b}{a}\right)^2 = \frac{b^2}{a^2} \]

Setting this equal to the product of the roots:

\[ \frac{b^2}{a^2} = \frac{5}{a} \]

Multiplying both sides by \( a^2 \):

\[ b^2 = 5a \]

### Step 3: Analyze the second equation

From the second equation \( x^2 - 2bx - 10 = 0 \):

- The sum of the roots \( \alpha + \beta = 2b \).
- The product of the roots \( \alpha \beta = -10 \).

### Step 4: Express \( \beta \) in terms of \( \alpha \)

From the product of the roots:

\[ \beta = \frac{-10}{\alpha} \]

### Step 5: Find \( \alpha^2 + \beta^2 \)

Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \):

1. Substitute \( \alpha + \beta = 2b \) and \( \alpha \beta = -10 \):

\[ \alpha^2 + \beta^2 = (2b)^2 - 2(-10) = 4b^2 + 20 \]

### Step 6: Substitute \( b^2 \)

From our earlier result \( b^2 = 5a \):

\[ \alpha^2 + \beta^2 = 4(5a) + 20 = 20a + 20 \]

### Step 7: Find \( a \)

We need to find the value of \( a \). From the equation we derived earlier:

\[ b^2 = 5a \implies b = \sqrt{5a} \]

Substituting \( b \) back into the equation \( 20a + 20 \):

1. We still need to express \( a \) in a solvable form. We can use the fact that \( b^2 = 5a \) to find \( a \) and \( b \).

### Step 8: Solve for \( a \) and \( b \)

Using the equations derived, we can solve for specific values of \( a \) and \( b \).

Assuming \( a = \frac{1}{4} \):

\[ b^2 = 5 \cdot \frac{1}{4} = \frac{5}{4} \implies b = \frac{\sqrt{5}}{2} \]

### Step 9: Calculate \( \alpha^2 + \beta^2 \)

Substituting \( a \) back into \( \alpha^2 + \beta^2 \):

\[ \alpha^2 + \beta^2 = 20 \cdot \frac{1}{4} + 20 = 5 + 20 = 25 \]

Thus, the final answer is:

\[ \alpha^2 + \beta^2 = 25 \]