Product of real roots of the equation `t^2x^2+|x|+9=0` a. is always +ve b. is always-ve c. does not exist d. none of these

Options

A. is always positive
B. is always negative
C. does not exist
D. none of these

Answer

To solve the equation \( t^2 x^2 + |x| + 9 = 0 \) and determine the product of its real roots, we will analyze the components of the equation step by step.

### Step 1: Analyze the components of the equation

The equation is given as:
\[ t^2 x^2 + |x| + 9 = 0 \]

Here, we have three terms:
1. \( t^2 x^2 \)
2. \( |x| \)
3. \( 9 \)

### Step 2: Determine the nature of each term

- The term \( t^2 x^2 \) is always non-negative because \( t^2 \) (the square of any real number) is non-negative and \( x^2 \) (the square of any real number) is also non-negative. Thus, \( t^2 x^2 \geq 0 \).

- The term \( |x| \) (the absolute value of \( x \)) is also always non-negative, meaning \( |x| \geq 0 \).

- The constant \( 9 \) is a positive number, so \( 9 > 0 \).

### Step 3: Combine the inequalities

Since all three terms are non-negative, we can conclude:
\[ t^2 x^2 + |x| + 9 \geq 0 + 0 + 9 = 9 \]

This means:
\[ t^2 x^2 + |x| + 9 > 0 \]

### Step 4: Conclusion about the roots

Since the expression \( t^2 x^2 + |x| + 9 \) is always greater than zero, it can never equal zero. Therefore, there are no real values of \( x \) that satisfy the equation:
\[ t^2 x^2 + |x| + 9 = 0 \]

### Step 5: Determine the product of the real roots

Since there are no real roots, the product of the real roots does not exist.

### Final Answer

Thus, the correct option is:
**c. does not exist**

---