If `ax^(2)+bx+c=0` and `5x^(2)+6x+12=0` have a common root where a, b and c are sides of a triangle `ABC`, then

Options

A. `Delta ABC` is obtuse angled
B. `Delta ABC` is acute angled
C. `Delta ABC` is right angled
D. no such triangle exists

Explanation

To solve the problem step by step, we need to analyze the given quadratic equations and the conditions for the sides of a triangle.

### Step 1: Identify the Quadratic Equations
We have two quadratic equations:
1. \( ax^2 + bx + c = 0 \)
2. \( 5x^2 + 6x + 12 = 0 \)

### Step 2: Find the Discriminant of the Second Equation
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
For the second equation \( 5x^2 + 6x + 12 = 0 \):
- \( A = 5 \)
- \( B = 6 \)
- \( C = 12 \)

Calculating the discriminant:
\[ D = 6^2 - 4 \cdot 5 \cdot 12 = 36 - 240 = -204 \]

### Step 3: Analyze the Discriminant
Since the discriminant \( D = -204 \) is negative, this means that the second equation has imaginary roots.

### Step 4: Common Root Condition
Since both equations have a common root, and the second equation has imaginary roots, it implies that the first equation must also have the same imaginary roots.

### Step 5: Set Up the Ratio Condition
For both quadratic equations to have a common root, the coefficients must be in proportion:
\[ \frac{a}{5} = \frac{b}{6} = \frac{c}{12} \]
Let this common ratio be \( k \). Therefore, we can express \( a \), \( b \), and \( c \) as:
\[ a = 5k, \quad b = 6k, \quad c = 12k \]

### Step 6: Check the Triangle Inequality
For \( a \), \( b \), and \( c \) to be the sides of a triangle, they must satisfy the triangle inequality:
1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Let's check the first condition:
\[ a + b = 5k + 6k = 11k \]
\[ c = 12k \]
Now, check if \( a + b > c \):
\[ 11k > 12k \]
This simplifies to:
\[ 11 > 12 \]
This is false.

### Step 7: Conclusion
Since the triangle inequality is not satisfied, \( a \), \( b \), and \( c \) cannot form a triangle. Therefore, the conclusion is that no such triangle exists.

### Final Answer
No such triangle exists.

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