If `a ,b ,c` be the sides of ` A B C` and equations `a x 62+b x+c=0a n d5x^2+12+13=0` have a common root, then find `/_Cdot`

Explanation

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Identify the equations
We have two quadratic equations:
1. \( ax^2 + bx + c = 0 \)
2. \( 5x^2 + 12x + 13 = 0 \)

These two equations have a common root.

### 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 \( 5x^2 + 12x + 13 = 0 \):
- \( A = 5 \)
- \( B = 12 \)
- \( C = 13 \)

Calculating the discriminant:
\[ D = 12^2 - 4 \cdot 5 \cdot 13 = 144 - 260 = -116 \]
Since the discriminant is negative, the roots of this equation are complex.

### Step 3: Understand the implications of complex roots
If one root of the second equation is \( p + i q \) (where \( i \) is the imaginary unit), the other root must be \( p - i q \). If the first equation also has a common root with the second equation, both roots of the first equation must also be \( p + i q \) and \( p - i q \).

### Step 4: Set up the ratio of coefficients
Since both quadratic equations have both roots in common, the coefficients of the equations must be in proportion:
\[ \frac{a}{5} = \frac{b}{12} = \frac{c}{13} \]
Let this common ratio be \( k \):
- \( a = 5k \)
- \( b = 12k \)
- \( c = 13k \)

### Step 5: Verify the triangle inequality
We need to check if \( a, b, c \) can form a triangle. The sides of the triangle are \( 5k, 12k, \) and \( 13k \).

### Step 6: Check if it forms a right triangle
Using the Pythagorean theorem:
\[ (5k)^2 + (12k)^2 = (13k)^2 \]
Calculating:
\[ 25k^2 + 144k^2 = 169k^2 \]
This holds true, indicating that the triangle is a right triangle.

### Step 7: Determine angle \( C \)
In a right triangle, the angle opposite the longest side (hypotenuse) is \( 90^\circ \). Since \( c = 13k \) is the longest side, angle \( C \) is:
\[ \angle C = 90^\circ \]

### Final Answer
Thus, the angle \( C \) is \( 90^\circ \).

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