Statement -1 : If a, b,c, `theta in R`, then the equation `a sin theta + b cos theta=c ` has a solution only if `c^2 gt a^2 + b^2 `
Statement-2 : If a,b,c, `theta ne R,` thent he minimuma nd maximum values of a `sintheta + b cos theta ` are respectively `-sqrt(a^2-b^2) and sqrt(a^2 +b^2)`

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True and 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

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
The statement claims that the equation \( a \sin \theta + b \cos \theta = c \) has a solution only if \( c^2 > a^2 + b^2 \).

1. **Start with the equation**:
\[ a \sin \theta + b \cos \theta = c \]

2. **Rearranging the equation**:
We can express this equation in a different form by dividing both sides by \( \sqrt{a^2 + b^2} \):
\[ \frac{a}{\sqrt{a^2 + b^2}} \sin \theta + \frac{b}{\sqrt{a^2 + b^2}} \cos \theta = \frac{c}{\sqrt{a^2 + b^2}} \]

3. **Let \( \cos \phi = \frac{a}{\sqrt{a^2 + b^2}} \) and \( \sin \phi = \frac{b}{\sqrt{a^2 + b^2}} \)**:
This implies that \( \cos^2 \phi + \sin^2 \phi = 1 \).

4. **Using the sine addition formula**:
The left-hand side can be rewritten using the sine addition formula:
\[ \sqrt{a^2 + b^2} \sin(\theta + \phi) = c \]

5. **Finding the range of \( \sin(\theta + \phi) \)**:
Since the sine function ranges from -1 to 1, we have:
\[ -\sqrt{a^2 + b^2} \leq c \leq \sqrt{a^2 + b^2} \]

6. **Conclusion for Statement 1**:
This implies that for the equation to have a solution, it must be true that:
\[ |c| \leq \sqrt{a^2 + b^2} \]
Therefore, the statement \( c^2 > a^2 + b^2 \) is **false**.

### Step 2: Analyze Statement 2
The statement claims that the minimum and maximum values of \( a \sin \theta + b \cos \theta \) are \( -\sqrt{a^2 + b^2} \) and \( \sqrt{a^2 + b^2} \) respectively.

1. **Using the result from Statement 1**:
From the analysis above, we have established that the maximum value of \( a \sin \theta + b \cos \theta \) is \( \sqrt{a^2 + b^2} \) and the minimum value is \( -\sqrt{a^2 + b^2} \).

2. **Conclusion for Statement 2**:
Therefore, Statement 2 is **true**.

### Final Conclusion
- Statement 1 is **false**.
- Statement 2 is **true**.

Thus, the correct answer is that Statement 1 is false and Statement 2 is true.