<p>The points (2, 5) and (5, 1) are two opposite vertices of a rectangle. If other two vertices are points on the straight line \(y = 2x + k\), then the value of \(k\) is:</p>
Step-by-Step Solution
Key Concept: The center of the rectangle is the midpoint of any diagonal. The other two vertices, being equidistant from this center, must lie on a line perpendicular to the diagonal connecting the given opposite vertices.
<p><strong>Step 1:</strong> Find the center (midpoint of diagonal) from (2,5) and (5,1):<br/>Center = ((2+5)/2, (5+1)/2) = (7/2, 3)</p><p><strong>Step 2:</strong> The slope of diagonal from (2,5) to (5,1) is:<br/>m₁ = (1-5)/(5-2) = -4/3</p><p><strong>Step 3:</strong> For a rectangle, the other diagonal (connecting the other two opposite vertices) must be perpendicular to this diagonal.<br/>Slope of other diagonal: m₂ = 3/4 (negative reciprocal of -4/3)</p><p><strong>Step 4:</strong> But the other two vertices lie on y = 2x + k, which has slope 2.<br/>This is NOT the diagonal slope. Instead, these two vertices and the center must satisfy: the line y = 2x + k passes through points equidistant from center (7/2, 3).</p><p><strong>Step 5:</strong> The center (7/2, 3) must lie on the perpendicular bisector property. For a rectangle, if two opposite vertices are (2,5) and (5,1), and the other two lie on y = 2x + k, then the center must satisfy the constraint that it's equidistant from all vertices.<br/>Substituting center (7/2, 3) into y = 2x + k:<br/>3 = 2(7/2) + k<br/>3 = 7 + k<br/>k = -4</p><p>∴ Answer: <strong>k = -4</strong> (Option C)</p>
Correct Answer: C