Equation of the circumcircle of a triangle formed by the lines `L_(1)=0,L_(2)=0andL_(3)=0` can be written as `L_(1)L_(2)+lamdaL_(2)L_(3)+muL_(3)L_(1)=0`, where `lamdaandmu` are such that coefficient of `x^(2)` =coefficient of `y^(2)` and coefficient of xy=0.
`L_(1)L_(2)^(2)+lamdaL_(2)L_(3)^(2)+muL_(1)^(2)=0` represents

Options

A. `lamdaL_(1)L_(4)+muL_(2)L_(3)=0`
B. `lamdaL_(1)L_(3)+muL_(2)L_(4)=0`
C. `lamdaL_(1)L_(2)+muL_(3)L_(4)=0`
D. `lamdaL_(1)^(2)L_(3)+muL_(2)^(2)L_(4)=0`

Answer

To solve the problem, we need to analyze the given equations and derive the required representation step by step.

### Step 1: Understand the Given Lines
The lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \) represent three lines in the plane. The intersection of these lines forms a triangle, and the circumcircle of this triangle can be expressed in terms of these lines.

**Hint:** Identify what the lines represent geometrically and how they interact.

### Step 2: Write the Equation of the Circumcircle
The circumcircle of the triangle formed by these lines can be expressed as:
\[ L_1 L_2 + \lambda L_2 L_3 + \mu L_3 L_1 = 0 \]
where \( \lambda \) and \( \mu \) are coefficients that need to satisfy certain conditions.

**Hint:** Recognize that the circumcircle equation is a quadratic in terms of \( x \) and \( y \).

### Step 3: Coefficient Conditions
The problem states that the coefficient of \( x^2 \) must equal the coefficient of \( y^2 \), and the coefficient of \( xy \) must be zero. This gives us two conditions to satisfy:
1. Coefficient of \( x^2 = \) Coefficient of \( y^2 \)
2. Coefficient of \( xy = 0 \)

**Hint:** Set up equations based on the coefficients of the terms in the circumcircle equation.

### Step 4: Analyze the Given Expression
The expression \( L_1 L_2^2 + \lambda L_2 L_3^2 + \mu L_1^2 = 0 \) needs to be analyzed to determine the type of curve it represents.

**Hint:** Substitute the lines into the expression and simplify.

### Step 5: Substitute Points
Let’s take a point \( (h, k) \) that satisfies \( L_1(h, k) = 0 \), \( L_2(h, k) = 0 \), and \( L_3(h, k) = 0 \). Substitute this point into the equation:
\[ L_1(h, k) L_2(h, k)^2 + \lambda L_2(h, k) L_3(h, k)^2 + \mu L_1(h, k)^2 = 0 \]
Since each \( L_i(h, k) = 0 \), the entire expression equals zero.

**Hint:** Understand that substituting the point will help in confirming the nature of the curve.

### Step 6: Conclusion
The expression \( L_1 L_2^2 + \lambda L_2 L_3^2 + \mu L_1^2 = 0 \) represents a curve that passes through the points of intersection of the lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \). This indicates that the curve is defined by the relationships between these lines.

**Final Answer:** The expression represents a curve passing through the point of intersection of the lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \).