Q)Two lines represented by the equation:

ax⁴ + bx³y +cx²y² +dxy³+ay⁴=0

will bisect the angles between the other two,if

a)c+6a=0 and ab+d=0

b)b+d=0 and a+c=0

c)c+6a=0 and b+d=0

d)None

To determine the conditions under which two lines represented by the equation \( ax^4 + bx^3y + cx^2y^2 + dxy^3 + ay^4 = 0 \) bisect the angles between two other lines, we need to delve into the properties of conic sections and the conditions for angle bisectors. The equation represents a quartic curve, and the lines can be derived from it under specific conditions.

Understanding the Angle Bisector Condition

For two lines to bisect the angles between two other lines, certain algebraic conditions must be satisfied. The general form of the angle bisector condition can be derived from the properties of the coefficients in the polynomial equation. Specifically, we need to analyze the relationships between the coefficients \( a, b, c, d \) in the given equation.

Analyzing the Given Options

Let's examine the provided options one by one:

• Option a: \( c + 6a = 0 \) and \( ab + d = 0 \)
• Option b: \( b + d = 0 \) and \( a + c = 0 \)
• Option c: \( c + 6a = 0 \) and \( b + d = 0 \)
• Option d: None

Exploring Each Condition

To find the correct conditions, we can use the relationships derived from the angle bisector theorem. The conditions typically involve the coefficients of the polynomial being related in a specific way.

1. **For Option a**: If \( c + 6a = 0 \), then \( c = -6a \). The second condition \( ab + d = 0 \) implies \( d = -ab \). This set of conditions can potentially satisfy the angle bisector requirement, but we need to verify further.

2. **For Option b**: The condition \( b + d = 0 \) gives \( d = -b \), and \( a + c = 0 \) leads to \( c = -a \). This combination does not necessarily guarantee that the lines will bisect the angles between two other lines.

3. **For Option c**: Here, \( c + 6a = 0 \) gives \( c = -6a \) and \( b + d = 0 \) gives \( d = -b \). This combination appears promising, as it maintains a relationship between the coefficients that could satisfy the angle bisector condition.

4. **For Option d**: This option suggests that none of the conditions are valid, which we can rule out if we find valid conditions in the previous options.

Conclusion on the Correct Option

After analyzing the conditions, it appears that Option c, \( c + 6a = 0 \) and \( b + d = 0 \), is the most likely to satisfy the angle bisector condition. This is because it maintains a consistent relationship among the coefficients that aligns with the geometric properties of angle bisectors in conic sections.

Therefore, the answer is Option c.