<p>Find the value of <i>k</i> if the mid-point of <i>A</i>(2, 5) and <i>B</i>(5, 1) lies on the line <i>y</i> = 2<i>x</i> + <i>k</i>.</p>
Step-by-Step Solution
Key Concept: Use the mid-point formula and substitute the mid-point coordinates into the line equation to find the parameter.
<p><strong>Step 1:</strong> Find the mid-point of A(2, 5) and B(5, 1):</p><p>$$C = \left(\frac{2+5}{2}, \frac{5+1}{2}\right) = \left(\frac{7}{2}, 3\right)$$</p><p><strong>Step 2:</strong> Since C lies on the line $y = 2x + k$, substitute the coordinates:</p><p>$$3 = 2 \cdot \frac{7}{2} + k$$</p><p>$$3 = 7 + k$$</p><p>$$k = -4$$</p>
Correct Answer: -4