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Difficulty: HardCoordinate Geometry and Lines

In the xyxy-plane, line L1L_1 passes through the points (2,5)(2, 5) and (6,−3)(6, -3). Line L2L_2 is perpendicular to line L1L_1 and passes through the point (1,2)(1, 2). Line L3L_3 is parallel to line L2L_2 and has a yy-intercept that is 55 units greater than the yy-intercept of line L2L_2. What is the xx-intercept of line L3L_3?

  1. −13-13Answer
  2. B
    −52-\frac{5}{2}
  3. C
    92\frac{9}{2}
  4. D
    1313
  5. E
    1515

Answer

The xx-intercept of line L3L_3 is −13-13.
The slope of line L1L_1 is −3−56−2=−2\frac{-3 - 5}{6 - 2} = -2. The negative reciprocal slope for line L2L_2 is 12\frac{1}{2}. Using point-slope form with point (1,2)(1,2), line L2L_2 is y=12x+32y = \frac{1}{2}x + \frac{3}{2}, giving a yy-intercept of 32\frac{3}{2}. Line L3L_3, being parallel to L2L_2, shares slope 12\frac{1}{2} and has yy-intercept 32+5=132\frac{3}{2} + 5 = \frac{13}{2}. Finding the xx-intercept of y=12x+132y = \frac{1}{2}x + \frac{13}{2} by setting y=0y=0 yields x=−13x = -13.

Step-by-Step Solution

1
Calculate the slope of line L1L_1
The slope m1=−3−56−2=−84=−2m_1 = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2.
The slope formula is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Determine the slope and equation of perpendicular line L2L_2
The slope m2=−1−2=12m_2 = -\frac{1}{-2} = \frac{1}{2}. Since L2L_2 passes through (1,2)(1, 2), its equation is y−2=12(x−1)y - 2 = \frac{1}{2}(x - 1), which simplifies to y=12x+32y = \frac{1}{2}x + \frac{3}{2}.
Perpendicular lines have negative reciprocal slopes.
3
Determine the equation of line L3L_3
The yy-intercept of L2L_2 is 32\frac{3}{2}. The yy-intercept of L3L_3 is 32+5=132\frac{3}{2} + 5 = \frac{13}{2}. Since L3L_3 is parallel to L2L_2, its slope is m3=12m_3 = \frac{1}{2}. Thus, the equation of L3L_3 is y=12x+132y = \frac{1}{2}x + \frac{13}{2}.
Parallel lines have equal slopes.
4
Find the xx-intercept of line L3L_3
Setting y=0y = 0 gives 0=12x+132  ⟹  12x=−132  ⟹  x=−130 = \frac{1}{2}x + \frac{13}{2} \implies \frac{1}{2}x = -\frac{13}{2} \implies x = -13.
The xx-intercept is the value of xx when y=0y = 0.

Key Concept

Slope of parallel and perpendicular lines, line equations, and intercept calculations
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Topics

Coordinate Geometry and Lines

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