If `sin alpha and cos alpha` are the roots of `ax^(2) + bx + c = 0` , then find the relation satisfied by a, b and c .

Answer

To find the relation satisfied by \( a, b, \) and \( c \) when \( \sin \alpha \) and \( \cos \alpha \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the roots
Let the roots of the quadratic equation be \( r_1 = \sin \alpha \) and \( r_2 = \cos \alpha \).

### Step 2: Use the sum and product of roots
From Vieta's formulas, we know:
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)

Substituting the values of the roots, we have:
1. \( \sin \alpha + \cos \alpha = -\frac{b}{a} \)
2. \( \sin \alpha \cdot \cos \alpha = \frac{c}{a} \)

### Step 3: Use the Pythagorean identity
From trigonometric identities, we know:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

### Step 4: Express \( \sin^2 \alpha + \cos^2 \alpha \) in terms of the roots
We can express \( \sin^2 \alpha + \cos^2 \alpha \) as follows:
\[ \sin^2 \alpha + \cos^2 \alpha = (\sin \alpha + \cos \alpha)^2 - 2\sin \alpha \cos \alpha \]

### Step 5: Substitute the expressions from Vieta's formulas
Substituting the values from Vieta's formulas into the identity:
\[ 1 = \left(-\frac{b}{a}\right)^2 - 2 \left(\frac{c}{a}\right) \]

### Step 6: Simplify the equation
Expanding and simplifying gives:
\[ 1 = \frac{b^2}{a^2} - \frac{2c}{a} \]

### Step 7: Multiply through by \( a^2 \)
To eliminate the denominators, multiply through by \( a^2 \):
\[ a^2 = b^2 - 2ac \]

### Step 8: Rearranging the equation
Rearranging gives us the final relation:
\[ b^2 - 2ac - a^2 = 0 \]

### Final Answer
The relation satisfied by \( a, b, \) and \( c \) is:
\[ b^2 - 2ac - a^2 = 0 \]