<p>(A) <i>y</i> = <i>x</i> + <i>a</i></p>
<p>(B) 2<i>y</i> = <i>a</i><i>x</i></p>
<p>(C) <i>y</i> = <i>ax</i></p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: A line passing through the origin with slope a has the equation y = ax. This is the standard form for lines through the origin, derived from the two-point form using points (0,0) and (1,a).
<p><strong>Step 1:</strong> Identify the general form of a straight line: y = mx + c, where m is slope and c is y-intercept.</p><p><strong>Step 2:</strong> For a line passing through the origin (0, 0), substitute the point into the general equation: 0 = m(0) + c, which gives c = 0.</p><p><strong>Step 3:</strong> The equation reduces to y = mx. If the slope is a, then the equation becomes y = ax.</p><p><strong>Step 4:</strong> Verify with option (A): y = x + a has y-intercept a ≠ 0, so it doesn't pass through origin unless a = 0.</p><p><strong>Step 5:</strong> Option (B): 2y = ax can be written as y = (a/2)x, which is valid but unnecessarily complicated compared to y = ax.</p><p><strong>Step 6:</strong> Option (C): y = ax is the standard, simplest form of a line through the origin with slope a.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C