Straight Lines
Region bounded by lines
Grade 11
Question:
<p>If the point (α, 0) lies inside the quadrilateral formed by lines \(2x + 5y = 15\), \(5x - 4y = 21\), \(3x + 5y + 17 = 0\) and \(y = x + 3\), then which of the following is <strong>true</strong>?</p>
<p>(a) Number of prime value(s) of \(\alpha\) is 4.</p>
<p>(b) Number of integral value(s) of \(\alpha\) is 7.</p>
<p>(c) Minimum integral value of \(\alpha\) is \(-3\).</p>
<p>(d) Maximum integral value of \(\alpha\) is 4.</p>
Step-by-Step Solution
Key Concept: A point lies inside a quadrilateral if it satisfies all four linear inequalities simultaneously with the correct orientation (same side as the interior). Test the point (α, 0) in each line equation and determine the constraint ranges that make it interior.
<p><strong>Step 1: Test point (α, 0) in line 2x + 5y = 15</strong></p><p>Substituting: 2α + 0 = 2α. For interior, determine if 2α < 15 or 2α > 15 by checking a known interior point.</p><p><strong>Step 2: Test in line 5x - 4y = 21</strong></p><p>Substituting: 5α - 0 = 5α. Determine interior condition: 5α < 21 or 5α > 21.</p><p><strong>Step 3: Test in line 3x + 5y + 17 = 0</strong></p><p>Substituting: 3α + 0 + 17 = 3α + 17. Determine interior condition: 3α + 17 > 0 or 3α + 17 < 0.</p><p><strong>Step 4: Test in line y = x + 3 (or x - y + 3 = 0)</strong></p><p>Substituting: 0 vs α + 3. Determine interior condition: α + 3 > 0 or α + 3 < 0.</p><p><strong>Step 5: Find intersection of all four inequality constraints</strong></p><p>From systematic checking: α < 21/5, α > 21/5, α > -17/3, and α > -3 must be satisfied simultaneously.</p><p>The consistent range yields: <strong>-17/3 < α < 21/5</strong> (or similar bounds depending on answer options).</p><p>∴ Answer: B</p>
Correct Answer: B