Statement-1:If the line y=x+c intersects the circle `x^2+y^2=r^2` in two real distinct points then `-rsqrt2 lt c lt r sqrt2`
Statement -2: If two circles intersects at two distinct points then `C_1C_2 lt r_1+r_2`

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

Answer

To solve the problem, we need to analyze the two statements provided and verify their validity.

### Step 1: Analyze Statement 1
The first statement claims that if the line \( y = x + c \) intersects the circle \( x^2 + y^2 = r^2 \) at two distinct points, then \( -r\sqrt{2} < c < r\sqrt{2} \).

1. **Substitute the line equation into the circle equation**:
\[ x^2 + (x + c)^2 = r^2 \]
Expanding this:
\[ x^2 + (x^2 + 2cx + c^2) = r^2 \]
\[ 2x^2 + 2cx + (c^2 - r^2) = 0 \]

2. **Determine the condition for two distinct points**:
For the quadratic equation \( 2x^2 + 2cx + (c^2 - r^2) = 0 \) to have two distinct real roots, the discriminant must be greater than zero:
\[ D = (2c)^2 - 4 \cdot 2 \cdot (c^2 - r^2) > 0 \]
Simplifying the discriminant:
\[ 4c^2 - 8(c^2 - r^2) > 0 \]
\[ 4c^2 - 8c^2 + 8r^2 > 0 \]
\[ -4c^2 + 8r^2 > 0 \]
\[ 8r^2 > 4c^2 \]
\[ 2r^2 > c^2 \]
Taking square roots:
\[ -\sqrt{2}r < c < \sqrt{2}r \]

### Conclusion for Statement 1:
Thus, Statement 1 is true: \( -r\sqrt{2} < c < r\sqrt{2} \).

### Step 2: Analyze Statement 2
The second statement claims that if two circles intersect at two distinct points, then the distance between their centers \( C_1C_2 < r_1 + r_2 \).

1. **Understanding the condition**:
For two circles to intersect at two distinct points, the distance between their centers must be less than the sum of their radii. This is a well-known geometric property.

2. **Using the triangle inequality**:
If \( C_1 \) and \( C_2 \) are the centers of the circles with radii \( r_1 \) and \( r_2 \), respectively, the condition for intersection is:
\[ C_1C_2 < r_1 + r_2 \]

### Conclusion for Statement 2:
Thus, Statement 2 is also true.

### Final Conclusion:
Both statements are true, but Statement 2 does not serve as a correct explanation for Statement 1. Therefore, the correct answer is that both statements are true, but Statement 2 is not a correct explanation of Statement 1.