Straight Lines
Angle between lines / region
Grade 11
Question:
<p>If \((\alpha, \alpha^2)\) falls inside the angle made by the lines \(2y = x,\ x > 0\) and \(y = 3x,\ x > 0\) then the set of values of \(\alpha\) is</p>
<p>A. \((-\alpha, 3)\)</p>
<p>B. \(\left(\dfrac{1}{2}, 3\right)\)</p>
<p>C. \((0, 3)\)</p>
<p>D. \((-\infty, 0) \cup \left(\dfrac{1}{2}, \infty\right)\)</p>
Step-by-Step Solution
Key Concept: A point lies inside the angle formed by two lines through the origin if it satisfies opposite inequality conditions with respect to both lines. For lines y = x/2 and y = 3x in the first quadrant, the point (α, α²) must satisfy x/2 < y < 3x simultaneously.
<p><strong>Step 1:</strong> Identify the two lines: y = x/2 (line 1) and y = 3x (line 2), both with x > 0.</p><p><strong>Step 2:</strong> For point (α, α²) to lie inside the angle, it must satisfy: α²/α > 1/2 AND α²/α < 3, which gives α > 1/2 and α < 3.</p><p><strong>Step 3:</strong> From first inequality: α > 1/2. From second: α < 3.</p><p><strong>Step 4:</strong> Since the point must be in the region x > 0, we need α > 0. But the constraint α > 1/2 is already stronger.</p><p><strong>Step 5:</strong> Also note that for (α, α²) to be in the angular region between these lines (both through origin), we need α > 0 for the point to have positive x-coordinate.</p><p><strong>Step 6:</strong> Combining: 1/2 < α < 3.</p><p>∴ Answer: B</p>
Correct Answer: B