Family Of Lines

Step 1: Condition for common line in two quadratic equations The given equations represent pairs of lines. If they have one line in common, their determinant condition must be satisfied: Solving this determinant equation, we find the possible values of .
Step 2: Calculating the determinant Expanding along the third column, Solving the determinant, So, satisfies: Thus, the sum of all possible values of is .

We are given the following two equations of lines:
1.
2.
We are required to find the area of the triangle formed by these lines.
Step 1: The equation represents a straight line. Let us first rewrite it in the standard form of a line: Step 2: The equation represents a pair of lines, which can be factored to find the lines formed. To factor the equation: We can factor it as: So, the two lines represented by this equation are: 1. 2. , or
Step 3: We now have the three lines:
1.
2.
3.
Step 4: To find the area of the triangle formed by these lines, we can use the formula for the area of a triangle with three vertices as: The points of intersection of these lines give the vertices of the triangle.
- The intersection of and gives the point .
- The intersection of and gives the point .
- The intersection of and gives the point , or .
Step 5: The area of the triangle is given by the formula: Thus, the area of the triangle formed by the lines is .

Step 1: Finding the vertices of
The lines given are: 1. 2. 3. Solving for intersections: - Intersection of and : So, . - Intersection of and : So, . - Intersection of and : So, .
Step 2: Finding the centroid of tetrahedron
The centroid of a tetrahedron with vertices , , , and is given by: Substituting , , , : Thus, the centroid is:

Step 1: Understanding the given line families The two given equations represent families of lines: The common line must be present in both families, meaning it should be a linear combination of both given equations.
Step 2: Forming the equation of the common line To find the common line, we take the determinant of the coefficients: Solving this determinant, we obtain the equation of the common line: Step 3: Finding From solving the determinant, we get and , so: Thus, .