Straight Lines
Parallel and Coincident Lines
Grade 11
Question:
<p><strong>Question 86:</strong> <strong>Statement-1:</strong> If the system of equations $2x + 3y = a$ and $bx + 4y = 5$ has infinite solutions, then $a = \frac{15}{8}$, $b = \frac{8}{5}$.</p><p><strong>Statement-2:</strong> Straight lines $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: Infinite solutions require the coefficient ratios to be equal (lines are coincident), not just parallel. Parallel lines share only the slope ratio condition.
<p>For infinite solutions, the lines must be identical, not just parallel. This requires $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.</p><p>For the given system: $\frac{2}{b} = \frac{3}{4} = \frac{a}{5}$</p><p>From $\frac{3}{4} = \frac{a}{5}$: $a = \frac{15}{4}$ (not $\frac{15}{8}$). From $\frac{2}{b} = \frac{3}{4}$: $b = \frac{8}{3}$ (not $\frac{8}{5}$).</p><p>Statement-1 is false.</p><p>Statement-2 correctly defines parallel lines. Statement-2 is true.</p><p>The answer is (d).</p>
Correct Answer: d