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 .

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, .