If the two curves `ax^(2) +2hxy +by^(2) +2g x+2fy +c=0 and d x^(2) +2h' xy +b' y^(2)+2g' x +2f'y +c'=0 ` intersect in four concylic points, then

Options

A. `(a-b)/(h)=(a'-b')/(h')`
B. `(a+b)/(h) =(d+b')/(h')`
C. `h(a-b)=h'(a'+b')`
D. `h(a+b) =h'(a'-b')`

Answer

To solve the problem, we need to derive the condition for two curves to intersect at four concyclic points. The curves are given by the equations:

1. \( S_1: ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \)
2. \( S_2: d x^2 + 2h' xy + b' y^2 + 2g' x + 2f' y + c' = 0 \)

### Step 1: Understand the condition for concyclic points
For four points to be concyclic, the coefficients of the quadratic terms in the combined equation must satisfy certain conditions. Specifically, the coefficients of \(x^2\), \(y^2\), and \(xy\) must satisfy the condition for a circle.

### Step 2: Form the linear combination of the two curves
We can express the condition for the intersection of the two curves as a linear combination:

\[ S = S_1 + \lambda S_2 = 0 \]

This leads to:

\[ (ax^2 + 2hxy + by^2 + 2gx + 2fy + c) + \lambda (dx^2 + 2h'xy + b'y^2 + 2g'x + 2f'y + c') = 0 \]

### Step 3: Combine the coefficients
Combining the coefficients for \(x^2\), \(y^2\), and \(xy\):

- Coefficient of \(x^2\): \(a + \lambda d\)
- Coefficient of \(y^2\): \(b + \lambda b'\)
- Coefficient of \(xy\): \(2h + \lambda 2h'\)

The equation becomes:

\[ (a + \lambda d)x^2 + (b + \lambda b')y^2 + (2h + \lambda 2h')xy + (2g + \lambda 2g')x + (2f + \lambda 2f')y + (c + \lambda c') = 0 \]

### Step 4: Set conditions for a circle
For this to represent a circle, the following conditions must hold:

1. Coefficient of \(x^2 =\) Coefficient of \(y^2\)
2. Coefficient of \(xy = 0\)

From the first condition:

\[ a + \lambda d = b + \lambda b' \]

From the second condition:

\[ 2h + \lambda 2h' = 0 \]

### Step 5: Solve for \(\lambda\)
From the first condition, we can express \(\lambda\):

\[ \lambda (d - b') = b - a \implies \lambda = \frac{a - b}{b' - d} \]

From the second condition:

\[ \lambda = -\frac{h}{h'} \]

### Step 6: Equate the two expressions for \(\lambda\)
Setting the two expressions for \(\lambda\) equal gives:

\[ \frac{a - b}{b' - d} = -\frac{h}{h'} \]

Cross-multiplying leads to:

\[ (a - b)h' = -(b' - d)h \]

Rearranging gives us the desired condition:

\[ (a - b)h + (b' - d)h' = 0 \]

### Final Condition
Thus, the condition for the two curves to intersect at four concyclic points is:

\[ (a - b)h = (b' - d)h' \]