<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 diagonals of a rectangle bisect each other at the same point. The center of the rectangle lies on the line containing the other two vertices, and this center is equidistant from all four vertices. Alternatively, the sides of a rectangle are perpendicular, so if the other two vertices lie on y = 2x + k, the sides from (2,5) and (5,1) must be perpendicular to this line.
<p><strong>Step 1:</strong> Find the center of the rectangle using the midpoint of the diagonal formed by (2,5) and (5,1).</p><p>Center = ((2+5)/2, (5+1)/2) = (7/2, 3)</p><p><strong>Step 2:</strong> Since the diagonals of a rectangle bisect each other, the center must lie on the line containing the other two vertices.</p><p>Substitute (7/2, 3) into y = 2x + k:</p><p>3 = 2(7/2) + k</p><p>3 = 7 + k</p><p>k = -4</p><p><strong>Step 3:</strong> Verify: The other two vertices also form a diagonal whose midpoint is (7/2, 3), and this line y = 2x - 4 passes through this center point.</p><p>∴ Answer: k = -4 (Option C)</p>
Correct Answer: C