Straight Lines
Intercept bisection and perpendicular lines
Grade None
Question:
<p>Line \(L_1\) passes through \(P(1, 2)\) and the portion of \(L_1\) intercepted between the axes is bisected at \(P(1, 2)\). Line \(L\) is perpendicular to \(L_1\) and passes through \((-2, 1)\). The point of intersection of \(L_1\) and \(L\) is:</p>
<p>\(\left(\dfrac{4}{5}, \dfrac{12}{5}\right)\)</p>
<p>\(\left(\dfrac{3}{5}, \dfrac{11}{5}\right)\)</p>
<p>\(\left(\dfrac{2}{5}, \dfrac{9}{5}\right)\)</p>
<p>\(\left(\dfrac{1}{5}, \dfrac{7}{5}\right)\)</p>
Step-by-Step Solution
Key Concept: If a line's intercept between axes is bisected at point P(1,2), then P is the midpoint of the segment joining the x-intercept (a,0) and y-intercept (0,b). Use midpoint formula: (a/2, b/2) = (1,2) to find a=2, b=4, then find the perpendicular line and solve simultaneously.
<p><strong>Step 1: Find equation of L₁</strong></p><p>Since P(1,2) bisects the intercept between axes, if intercepts are A(a,0) and B(0,b):</p><p>Midpoint: (a/2, b/2) = (1, 2)</p><p>Therefore: a = 2, b = 4</p><p>Line L₁ passes through (2,0) and (0,4)</p><p><strong>Step 2: Write equation of L₁</strong></p><p>Using intercept form: x/2 + y/4 = 1</p><p>Or: 2x + y = 4</p><p>Slope of L₁: m₁ = -2</p><p><strong>Step 3: Find equation of L</strong></p><p>Since L ⊥ L₁: m₂ = 1/2 (negative reciprocal of -2)</p><p>L passes through (-2, 1) with slope 1/2:</p><p>y - 1 = (1/2)(x + 2)</p><p>2y - 2 = x + 2</p><p>x - 2y + 4 = 0</p><p><strong>Step 4: Find intersection point</strong></p><p>Solve simultaneously:</p><p>2x + y = 4 ... (1)</p><p>x - 2y + 4 = 0 ... (2)</p><p>From (2): x = 2y - 4</p><p>Substitute in (1): 2(2y - 4) + y = 4</p><p>4y - 8 + y = 4</p><p>5y = 12</p><p>y = 12/5</p><p>x = 2(12/5) - 4 = 24/5 - 20/5 = 4/5</p><p>∴ Answer: A (point is (4/5, 12/5))</p>
Correct Answer: A