If the quadratic equation `a x^2+b x+c=0(a >0)` has `sec^2thetaa n dcos e c^2theta` as its roots, then which of the following must hold good? `b+c=0` b. `b^2-4a cgeq0` c. `ccgeq4a` d. `4a+bgeq0`

Options

A. ` b + c = 0`
B. ` b^(2) - 4 a c ge 0 `
C. ` c ge 4a `
D. ` 4a b ge 0 `

Explanation

To solve the given problem, we need to analyze the quadratic equation \( ax^2 + bx + c = 0 \) where \( a > 0 \) and the roots are \( \sec^2 \theta \) and \( \csc^2 \theta \).

### Step 1: Sum and Product of Roots
The sum of the roots \( \sec^2 \theta + \csc^2 \theta \) can be expressed using the identity:
\[ \sec^2 \theta + \csc^2 \theta = \frac{1}{\cos^2 \theta} + \frac{1}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]
The product of the roots \( \sec^2 \theta \cdot \csc^2 \theta \) is given by:
\[ \sec^2 \theta \cdot \csc^2 \theta = \frac{1}{\cos^2 \theta} \cdot \frac{1}{\sin^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]

### Step 2: Relating to Coefficients
From Vieta's formulas, we know:
- The sum of the roots \( \sec^2 \theta + \csc^2 \theta = -\frac{b}{a} \)
- The product of the roots \( \sec^2 \theta \cdot \csc^2 \theta = \frac{c}{a} \)

### Step 3: Setting Up the Equations
From the above relationships, we can set up the following equations:
1. \( \sec^2 \theta + \csc^2 \theta = -\frac{b}{a} \)
2. \( \sec^2 \theta \cdot \csc^2 \theta = \frac{c}{a} \)

### Step 4: Finding \( b + c = 0 \)
From the sum of the roots, we have:
\[ -\frac{b}{a} = \sec^2 \theta + \csc^2 \theta \]
From the product of the roots:
\[ \frac{c}{a} = \sec^2 \theta \cdot \csc^2 \theta \]
We can express \( b \) and \( c \) in terms of \( a \):
- \( b = -a(\sec^2 \theta + \csc^2 \theta) \)
- \( c = a(\sec^2 \theta \cdot \csc^2 \theta) \)

Now, substituting these into the equation \( b + c = 0 \):
\[ -a(\sec^2 \theta + \csc^2 \theta) + a(\sec^2 \theta \cdot \csc^2 \theta) = 0 \]
This simplifies to:
\[ a(\sec^2 \theta \cdot \csc^2 \theta - (\sec^2 \theta + \csc^2 \theta)) = 0 \]
Since \( a > 0 \), we have:
\[ \sec^2 \theta \cdot \csc^2 \theta = \sec^2 \theta + \csc^2 \theta \]
Thus, \( b + c = 0 \) holds true.

### Step 5: Discriminant Condition
For the quadratic equation to have real roots, the discriminant must be non-negative:
\[ b^2 - 4ac \geq 0 \]
Substituting \( b = -c \) into the discriminant gives:
\[ (-c)^2 - 4ac \geq 0 \implies c^2 - 4ac \geq 0 \implies c(c - 4a) \geq 0 \]
This implies that either \( c \geq 0 \) and \( c - 4a \geq 0 \) or \( c \leq 0 \) and \( c - 4a \leq 0 \).

### Step 6: Analyzing \( c \geq 4a \)
From the previous step, we can conclude that:
\[ c \geq 4a \]
is a valid condition.

### Step 7: Evaluating \( 4a + b \geq 0 \)
Substituting \( b = -c \) into \( 4a + b \):
\[ 4a - c \geq 0 \implies c \leq 4a \]
This contradicts our previous result that \( c \geq 4a \). Thus, \( 4a + b \geq 0 \) does not hold.

### Conclusion
The conditions that must hold true are:
- \( b + c = 0 \)
- \( b^2 - 4ac \geq 0 \)
- \( c \geq 4a \)

Thus, the valid options are:
- **a. \( b + c = 0 \)**
- **b. \( b^2 - 4ac \geq 0 \)**
- **c. \( c \geq 4a \)**