Question

The Graph of the equations \(x + 3y - 2 = 0\) and \(2x - 5y + 1 = 0\) are :

• A. Coincident lines
• B. Parallel lines
• C. Perpendicular lines
• D. Intersecting lines

Hint:

For lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\): 1. Calculate \(\frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\). Equation 1: \(x + 3y - 2 = 0 \Rightarrow a_1=1, b_1=3\) Equation 2: \(2x - 5y + 1 = 0 \Rightarrow a_2=2, b_2=-5\) \(\frac{a_1}{a_2} = \frac{1}{2}\) \(\frac{b_1}{b_2} = \frac{3}{-5}\) 2. If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the lines are Intersecting. Here, \(\frac{1}{2} \neq \frac{3}{-5}\), so they are intersecting lines. (To check for parallel, you'd also need \(\frac{c_1}{c_2}\) if the first two ratios were equal. To check for coincident, all three ratios must be equal).

Answer 1

The Correct Option is D

Concept: For a pair of linear equations in two variables, \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the nature of their graphs (and solutions) can be determined by comparing the ratios of their coefficients: (A) If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the lines are intersecting (at exactly one point). (B) If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the lines are coincident (they are the same line). (C) If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), the lines are parallel and distinct (no intersection). For perpendicular lines, the product of their slopes is -1 (\(m_1 m_2 = -1\)). The slope \(m = -a/b\). So, \( (-\frac{a_1}{b_1})(-\frac{a_2}{b_2}) = -1 \Rightarrow a_1a_2 + b_1b_2 = 0 \). Step 1: Identify the coefficients for both equations Equation 1: \(x + 3y - 2 = 0\) \(a_1 = 1\), \(b_1 = 3\), \(c_1 = -2\) Equation 2: \(2x - 5y + 1 = 0\) \(a_2 = 2\), \(b_2 = -5\), \(c_2 = 1\) Step 2: Calculate the ratios \(\frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\) \[ \frac{a_1}{a_2} = \frac{1}{2} \] \[ \frac{b_1}{b_2} = \frac{3}{-5} = -\frac{3}{5} \] Step 3: Compare these two ratios We see that \(\frac{1}{2} \neq -\frac{3}{5}\). So, \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\). Step 4: Determine the nature of the lines Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the lines are intersecting lines. They will meet at exactly one point. Step 5: Check for perpendicularity (optional, as intersecting is already an option) For lines to be perpendicular, \(a_1a_2 + b_1b_2 = 0\). \( (1)(2) + (3)(-5) = 2 - 15 = -13 \). Since \(-13 \neq 0\), the lines are not perpendicular. The lines are intersecting (but not perpendicular). This matches option (4).