Statement 1: The quadratic polynomial `y=ax^(2)+bx+c(a!=0` and `a,b in R)` is symmetric about the line `2ax+b=0`
Statement 2: Parabola is symmetric about its axis of symmetry.

Options

A. Statement -1 is true, Statement -2 is true, Statement -2 is a correct explanation for Statement-1
B. Statement -1 is true, Statement -2 is true, Statement -2 is not a correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false, Statement -2 is true

Explanation

To solve the problem, we need to analyze the two statements given about the quadratic polynomial and the parabola.

### Step 1: Understanding the Quadratic Polynomial
The quadratic polynomial is given by:
\[ y = ax^2 + bx + c \]
where \( a \neq 0 \) and \( a, b, c \in \mathbb{R} \).

### Step 2: Finding the Axis of Symmetry
The axis of symmetry for a quadratic polynomial in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
This means that the parabola is symmetric about the vertical line \( x = -\frac{b}{2a} \).

### Step 3: Rewriting the Polynomial
We can rewrite the polynomial in a way that highlights its vertex. Completing the square:
\[ y = a\left(x^2 + \frac{b}{a}x\right) + c \]
To complete the square:
\[ = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \]
\[ = a\left(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \]
\[ = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c \]

### Step 4: Identifying the Vertex
The vertex of the parabola is at:
\[ \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right) \]
The axis of symmetry is the vertical line \( x = -\frac{b}{2a} \).

### Step 5: Analyzing Statement 1
Statement 1 claims that the quadratic polynomial is symmetric about the line \( 2ax + b = 0 \). Rearranging gives:
\[ x = -\frac{b}{2a} \]
This is indeed the axis of symmetry we found in Step 2. Thus, Statement 1 is **true**.

### Step 6: Analyzing Statement 2
Statement 2 states that a parabola is symmetric about its axis of symmetry. Since we have established that the axis of symmetry for the parabola defined by the quadratic polynomial is \( x = -\frac{b}{2a} \), it follows that the parabola is symmetric about this line. Thus, Statement 2 is also **true**.

### Conclusion
Both statements are true:
- Statement 1: True
- Statement 2: True