Question

Find the equation of a straight line that passes through the intersection of the lines $L_1: 3x + 4y + 1 = 0$ and $L_2: 2x + 3y - 7 = 0$ and satisfies one of the following conditions:

• A.
  1. It also passes through (1, 3)
• B.
  2. It cuts off equal intercepts on the axes
• C.
  3. Is parallel to y-axis
• D.
  4. Is parallel to x-axis
• E.
  5. Also passes through (2, 1)

Answer 1

Answer: (A), (B), (C), (D), (E)

The question asks to find the equation of a straight line for each of the given conditions. The general equation of a line passing through the intersection of $L_1=0$ and $L_2=0$ is $L_1 + \lambda L_2 = 0$.

1. Condition 1 (Passes through (1, 3)): Substituting (1, 3) into $(3x + 4y + 1) + \lambda(2x + 3y - 7) = 0$ gives $16 + 4\lambda = 0$, so $\lambda = -4$. The equation is $(3x + 4y + 1) - 4(2x + 3y - 7) = 0$, which simplifies to $5x + 8y - 29 = 0$.

2. Condition 2 (Cuts off equal intercepts): The equation is $(3 + 2\lambda)x + (4 + 3\lambda)y + (1 - 7\lambda) = 0$. For equal intercepts, either $1 - 7\lambda = 0$ (leading to $\lambda = 1/7$ and the line $23x + 31y = 0$) or $3 + 2\lambda = 4 + 3\lambda$ (leading to $\lambda = -1$ and the line $x + y + 8 = 0$). The primary answer for non-zero equal intercepts is $x + y + 8 = 0$.

3. Condition 3 (Parallel to y-axis): The coefficient of $y$ must be zero: $4 + 3\lambda = 0$, so $\lambda = -4/3$. The equation is $(3x + 4y + 1) - \frac{4}{3}(2x + 3y - 7) = 0$, which simplifies to $x + 31 = 0$.

4. Condition 4 (Parallel to x-axis): The coefficient of $x$ must be zero: $3 + 2\lambda = 0$, so $\lambda = -3/2$. The equation is $(3x + 4y + 1) - \frac{3}{2}(2x + 3y - 7) = 0$, which simplifies to $y - 23 = 0$.

5. Condition 5 (Passes through (2, 1)): The point (2, 1) lies on $L_2: 2x + 3y - 7 = 0$. Since the family of lines includes $L_2=0$, the equation is $2x + 3y - 7 = 0$.

Since all 5 conditions lead to a valid line, and the question asks to "Find the equation", the answer is the set of equations derived for each condition.

Answer 2

The family of straight lines passing through the intersection of two lines $L_1 = 0$ and $L_2 = 0$ is represented by the equation $L_1 + \lambda L_2 = 0$, where $\lambda$ is a real parameter. This parametric form allows us to find a specific line that satisfies additional conditions. Each condition provided in the question allows us to solve for a unique value of $\lambda$, thereby determining the equation of the specific line.

• Condition 1: Substituting the coordinates of the point (1, 3) into the general equation allows us to find $\lambda$.
• Condition 2: The condition of cutting off equal intercepts on the axes leads to two possible scenarios, one where the intercepts are zero and one where they are non-zero and equal. This provides two possible values for $\lambda$.
• Condition 3 & 4: A line parallel to the y-axis has an undefined slope (coefficient of y is zero), and a line parallel to the x-axis has a zero slope (coefficient of x is zero). Setting the respective coefficients in the general equation to zero allows us to solve for $\lambda$.
• Condition 5: If the given point lies on one of the original lines ($L_2$ in this case), then that original line itself is part of the family of lines and satisfies the condition. This corresponds to the case where $\lambda$ is infinite, or when the line is $L_2 = 0$.

By applying these principles, we derive the specific equation for each of the five conditions.