Straight Lines
Intersection of Lines
Grade 11
Question:
<p>Straight lines $2x + y = 5$ and $x - 2y = 3$ intersect at point A. Points B and C are chosen on these two lines such that $AB = AC$. Then the equation of a line BC passing through the point $(2, 3)$ is:</p>
<p>(a) $3x - y - 3 = 0$</p>
<p>(b) $x + 3y - 11 = 0$</p>
<p>(c) $3x + y - 9 = 0$</p>
<p>(d) $x - 3y + 7 = 0$</p>
Step-by-Step Solution
Key Concept: Since AB = AC, point A lies on the perpendicular bisector of segment BC. The line BC must be perpendicular to the angle bisector of the two given lines at their intersection point A.
<p><strong>Step 1:</strong> Find point A, the intersection of lines $2x + y = 5$ and $x - 2y = 3$.</p><p>From $2x + y = 5$: $y = 5 - 2x$</p><p>Substituting into $x - 2y = 3$: $x - 2(5 - 2x) = 3$</p><p>$x - 10 + 4x = 3 ⟹ 5x = 13 ⟹ x = \frac{13}{5}$</p><p>$y = 5 - 2(\frac{13}{5}) = 5 - \frac{26}{5} = -\frac{1}{5}$</p><p>So $A = (\frac{13}{5}, -\frac{1}{5})$</p><p><strong>Step 2:</strong> Find the slopes of the given lines.</p><p>Line 1: $2x + y = 5 ⟹ m_1 = -2$</p><p>Line 2: $x - 2y = 3 ⟹ m_2 = \frac{1}{2}$</p><p><strong>Step 3:</strong> Since AB = AC with B on line 1 and C on line 2, point A is equidistant from both lines. The line BC is perpendicular to the angle bisector of these two lines at A.</p><p>The angle bisectors have slopes found from: $\frac{m_1 - m}{1 + m_1 m} = -\frac{m - m_2}{1 + m \cdot m_2}$</p><p>For the angle bisectors: $\frac{-2 - m}{1 - 2m} = \pm \frac{m - \frac{1}{2}}{1 + \frac{m}{2}}$</p><p>Solving: $m = 3$ or $m = -\frac{1}{3}$</p><p><strong>Step 4:</strong> Line BC is perpendicular to one of the angle bisectors. If the angle bisector has slope 3, then BC has slope $-\frac{1}{3}$. If the angle bisector has slope $-\frac{1}{3}$, then BC has slope 3.</p><p><strong>Step 5:</strong> Check which line passes through $(2, 3)$:</p><p>For $x + 3y - 11 = 0$: $2 + 3(3) - 11 = 2 + 9 - 11 = 0$ ✓</p><p>This line has slope $-\frac{1}{3}$ (perpendicular to slope 3 angle bisector).</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B