If the pair of lines `ax^2+2(a+b)xy+by^2=0` lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then

Options

A. `3a^2+2ab+3b^2=0`
B. `3a^2+10ab+3b^2=0`
C. `3a^2-2ab+3b^2=0`
D. `3a^2-10ab+3b^2=0`

Answer

To solve the problem, we need to analyze the given equation of the pair of lines and the conditions provided regarding the sectors formed by these lines in a circle.

### Step-by-Step Solution:

1. **Understand the Equation of the Pair of Lines:**
The equation given is \( ax^2 + 2(a+b)xy + by^2 = 0 \). This represents a pair of straight lines through the origin.

2. **Identify the Angles of the Sectors:**
It is given that the area of one sector is thrice the area of another sector. If we denote the areas of the sectors as \( A \) and \( B \), we can express this relationship as:
\[ B = 3A \]
Since the total angle around a point is \( 360^\circ \), we can express the angles corresponding to the sectors:
\[ A + B = 180^\circ \]
Substituting \( B \) gives:
\[ A + 3A = 180^\circ \implies 4A = 180^\circ \implies A = 45^\circ \quad \text{and} \quad B = 135^\circ \]

3. **Relate the Angles to the Pair of Lines:**
The angle \( \theta \) between the two lines can be calculated using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
where \( h = a + b \) from the general form of the equation of the pair of lines.

4. **Set the Angle Equal to \( 45^\circ \):**
Since one of the sectors corresponds to an angle of \( 45^\circ \), we have:
\[ \tan 45^\circ = 1 \]
Therefore, we set up the equation:
\[ 1 = \frac{2\sqrt{(a+b)^2 - ab}}{a + b} \]

5. **Square Both Sides:**
Squaring both sides gives:
\[ 1 = \frac{4((a+b)^2 - ab)}{(a+b)^2} \]
This simplifies to:
\[ (a+b)^2 = 4((a+b)^2 - ab) \]

6. **Rearranging the Equation:**
Expanding and rearranging leads to:
\[ (a+b)^2 = 4(a^2 + 2ab + b^2 - ab) \implies (a+b)^2 = 4(a^2 + ab + b^2) \]

7. **Final Form:**
This results in:
\[ 4a^2 + 4b^2 + 8ab = a^2 + 2ab + b^2 \]
Rearranging gives:
\[ 3a^2 + 2ab + 3b^2 = 0 \]

### Conclusion:
The required condition for the coefficients \( a \) and \( b \) is:
\[ 3a^2 + 2ab + 3b^2 = 0 \]