Question

Which of the following is the equation of a family of lines passing through the point of intersection of the lines $3x - 4y + 6 = 0$ and $5x + 2y - 7 = 0$?

• A. $(3x + 4y + 6) + k(5x + 2y - 7) = 0$
• B. $(3x - 4) + k(5x + 2y) = 0$
• C. $(3x + 4) + k(5x + 2y - 7) = 0$
• D. $(3x - 4) + k(5x + 2 + 7) = 0$

Answer 1

Answer: (A)

$(3x + 4y + 6) + k(5x + 2y - 7) = 0$

Answer 2

The equation of a family of lines passing through the intersection of two lines $L_1=0$ and $L_2=0$ is $L_1 + kL_2 = 0$. For the given lines $3x - 4y + 6 = 0$ and $5x + 2y - 7 = 0$, the correct family equation is $(3x - 4y + 6) + k(5x + 2y - 7) = 0$. Among the given options, $(3x + 4y + 6) + k(5x + 2y - 7) = 0$ is the closest match, differing only by the sign of the $y$-term in the first part, likely due to a typographical error in the question's options.