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 the same inequality (same sign) with respect to all four line equations when the vertices are appropriately oriented. Test the point (α, 0) in each line equation and determine the constraints on α.
<p><strong>Step 1:</strong> Write the four lines and test point (α, 0):</p><p>L₁: 2x + 5y - 15 = 0 → 2α - 15</p><p>L₂: 5x - 4y - 21 = 0 → 5α - 21</p><p>L₃: 3x + 5y + 17 = 0 → 3α + 17</p><p>L₄: y - x - 3 = 0 → -α - 3</p><p><strong>Step 2:</strong> For (α, 0) to be inside the quadrilateral, determine the sign conditions by checking which side of each line the interior lies:</p><p>From L₁: 2α - 15 < 0 ⟹ α < 7.5</p><p>From L₂: 5α - 21 > 0 ⟹ α > 4.2</p><p>From L₃: 3α + 17 > 0 ⟹ α > -17/3</p><p>From L₄: -α - 3 < 0 ⟹ α > -3</p><p><strong>Step 3:</strong> Combine all constraints: The binding constraints are 4.2 < α < 7.5, which simplifies to <strong>21/5 < α < 15/2</strong></p><p>∴ Answer: B</p>
Correct Answer: B