Straight Lines
Concurrency of lines
Grade 11
Question:
<p>For what values of λ the following three lines are concurrent?<br>\(x + y = 1\), \(\lambda x + 2y = 3\), \(\lambda^2 x + 4y + 9 = 0\)</p>
<p>\(\lambda = -13 \pm \sqrt{15}\)</p>
<p>\(\lambda = \dfrac{-13 \pm \sqrt{289}}{2}\)</p>
<p>\(\lambda = 2, -15\)</p>
<p>\(\lambda = -2, 15\)</p>
Step-by-Step Solution
Key Concept: Three lines are concurrent if they all pass through a single point. Find the intersection of the first two lines, then substitute this point into the third line to find the value(s) of λ that satisfy all three equations simultaneously.
<p><strong>Step 1:</strong> Find intersection of first two lines.</p><p>From x + y = 1, we get y = 1 - x</p><p>Substitute into λx + 2y = 3:</p><p>λx + 2(1 - x) = 3</p><p>λx + 2 - 2x = 3</p><p>(λ - 2)x = 1</p><p>x = 1/(λ - 2) (assuming λ ≠ 2)</p><p>y = 1 - x = 1 - 1/(λ - 2) = (λ - 3)/(λ - 2)</p><p><strong>Step 2:</strong> For concurrency, this point must lie on the third line:</p><p>λ²x + 4y + 9 = 0</p><p>λ² · 1/(λ - 2) + 4 · (λ - 3)/(λ - 2) + 9 = 0</p><p>[λ² + 4(λ - 3) + 9(λ - 2)]/(λ - 2) = 0</p><p>λ² + 4λ - 12 + 9λ - 18 = 0</p><p>λ² + 13λ - 30 = 0</p><p><strong>Step 3:</strong> Factorize: (λ + 15)(λ - 2) = 0</p><p>λ = -15 or λ = 2</p><p><strong>Step 4:</strong> Check λ = 2: This makes denominator zero, so first two lines are parallel. Reject λ = 2.</p><p>∴ Answer: λ = -15</p>
Correct Answer: C