Statement 1`:` If two conics `a_(1)x^(2)+ 2h_(1)xy+b_(1)^(2)=c_(1) , a_(2)x^(2) +2h_(2) xy +b_(2)y^(2) =c_(2)` intersect in 4 concyclic points, then `( a_(1) -b_(1)) h_(2)=(a_(2)-b_(2))h_(1)`. Statement 2 `:` For a conic to be a circle, coefficient of `x^(2) =` coefficient of `y^(2)` and coefficient of xy =0.

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 problem, we need to prove the first statement regarding the intersection of two conics and their relationship when they intersect at four concyclic points.

### Step-by-Step Solution:

1. **Understanding the Conics**:
We have two conic equations:
\[ a_1 x^2 + 2h_1 xy + b_1 y^2 = c_1 \]
\[ a_2 x^2 + 2h_2 xy + b_2 y^2 = c_2 \]

2. **General Form of Conics**:
The general equation of a conic is given by:
\[ Ax^2 + By^2 + 2Gxy + 2Fx + 2Hy + C = 0 \]
Here, we can identify:
- \( A = a_1 \)
- \( B = b_1 \)
- \( G = h_1 \)
- \( C = -c_1 \)

And similarly for the second conic:
- \( A = a_2 \)
- \( B = b_2 \)
- \( G = h_2 \)
- \( C = -c_2 \)

3. **Condition for Concyclic Points**:
For the two conics to intersect at four concyclic points, we need to derive a relationship between their coefficients.

4. **Multiplying the Equations**:
We multiply the first conic by \( h_2 \) and the second conic by \( h_1 \):
\[ h_2(a_1 x^2 + 2h_1 xy + b_1 y^2) = h_2 c_1 \]
\[ h_1(a_2 x^2 + 2h_2 xy + b_2 y^2) = h_1 c_2 \]

5. **Rearranging the Equations**:
After multiplying, we can rewrite the equations:
\[ a_1 h_2 x^2 + 2h_1 h_2 xy + b_1 h_2 y^2 = h_2 c_1 \]
\[ a_2 h_1 x^2 + 2h_1 h_2 xy + b_2 h_1 y^2 = h_1 c_2 \]

6. **Subtracting the Equations**:
Now, we subtract the second equation from the first:
\[ (a_1 h_2 - a_2 h_1)x^2 + (b_1 h_2 - b_2 h_1)y^2 = h_2 c_1 - h_1 c_2 \]

7. **Setting Coefficients Equal**:
For the conics to have four concyclic points, the coefficients of \( x^2 \) and \( y^2 \) must be equal:
\[ a_1 h_2 - b_1 h_2 = a_2 h_1 - b_2 h_1 \]
Rearranging gives:
\[ (a_1 - b_1)h_2 = (a_2 - b_2)h_1 \]

8. **Conclusion**:
Thus, we have proved the statement:
\[ (a_1 - b_1) h_2 = (a_2 - b_2) h_1 \]