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.
If `L_(1)L_(2)+lamdaL_(2)L_(3)+muL_(3)L_(1)=0` is such that `mu=0andlamda` is non-zero, then it represents

Options

A. a parabola
B. a pair of straight lines
C. a circle
D. an ellipse

Answer

To solve the problem, we need to analyze the given equation of the circumcircle of a triangle formed by the lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \). The equation 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 we need to determine under the conditions that the coefficient of \( x^2 \) equals the coefficient of \( y^2 \) and the coefficient of \( xy = 0 \).

### Step 1: Understanding the Lines
The lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \) represent three lines in the Cartesian plane. The intersection points of these lines will form the vertices of the triangle.

### Step 2: Setting Up the Equation
The circumcircle of the triangle can be represented by the equation:

\[ L_1 L_2 + \lambda L_2 L_3 + \mu L_3 L_1 = 0 \]

### Step 3: Analyzing the Coefficients
We know that:
- The coefficient of \( x^2 \) is equal to the coefficient of \( y^2 \).
- The coefficient of \( xy = 0 \).

This means we need to ensure that the terms involving \( \lambda \) and \( \mu \) satisfy these conditions.

### Step 4: Substituting Values
Given that \( \mu = 0 \) and \( \lambda \) is non-zero, we can simplify the equation to:

\[ L_1 L_2 + \lambda L_2 L_3 = 0 \]

### Step 5: Evaluating the Equation
Now, substituting \( L_1, L_2, L_3 \) with their respective values (which are functions of \( x \) and \( y \)), we can analyze the behavior of the equation.

### Step 6: Conclusion
Since \( L_1, L_2, L_3 \) represent the lines that form the triangle, and the equation represents a curve, we can conclude that the equation represents a curve passing through the points of intersection of the lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \).

### Final Answer
Thus, the final answer is that the equation represents a curve passing through the points of intersection of the lines \( L_1 = 0 \), \( L_2 = 0 \), and \( L_3 = 0 \).

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