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Difficulty: HardCoordinate Geometry and Lines

In the xyxy-plane, line KK is defined by the equation 3x−4y=123x - 4y = 12. Line MM is perpendicular to line KK and passes through the point (6,−1)(6, -1). Which of the following statements regarding line MM must be true? Select all that apply.

  1. Line MM has a yy-intercept of (0,7)(0, 7).Answer
  2. Line MM passes through the point (−3,11)(-3, 11).Answer
  3. C
    Line MM has an xx-intercept of (7,0)(7, 0).
  4. D
    Line MM passes through Quadrant III.
  5. E
    The point of intersection between line KK and line MM lies in Quadrant IV.

Answer

The statements asserting that Line MM has a yy-intercept of (0,7)(0, 7) and that Line MM passes through the point (−3,11)(-3, 11) are correct.
Line KK has a slope of 34\frac{3}{4}, making the perpendicular slope of line MM equal to −43-\frac{4}{3}. Using the point (6,−1)(6, -1), the equation of line MM is y=−43x+7y = -\frac{4}{3}x + 7. Evaluating the options: setting x=0x = 0 gives y=7y = 7, confirming the yy-intercept is (0,7)(0, 7); substituting x=−3x = -3 yields y=11y = 11, confirming (−3,11)(-3, 11) lies on line MM. Both of these statements are true.

Step-by-Step Solution

1
Determine the slope of line KK and the perpendicular slope of line MM.
Line KK in slope-intercept form is y=34x−3y = \frac{3}{4}x - 3, so its slope is mK=34m_K = \frac{3}{4}. The perpendicular slope for line MM is the negative reciprocal: mM=−43m_M = -\frac{4}{3}.
Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.
2
Find the equation of line MM using point-slope form with point (6,−1)(6, -1).
y−(−1)=−43(x−6)  ⟹  y+1=−43x+8  ⟹  y=−43x+7y - (-1) = -\frac{4}{3}(x - 6) \implies y + 1 = -\frac{4}{3}x + 8 \implies y = -\frac{4}{3}x + 7.
Knowing the slope and a point on the line allows determination of the line's exact linear equation.
3
Evaluate the statements using the equation of line MM.
1. yy-intercept: set x=0  ⟹  y=7x = 0 \implies y = 7, so (0,7)(0,7) is correct.
2. Point (−3,11)(-3, 11): y=−43(−3)+7=4+7=11y = -\frac{4}{3}(-3) + 7 = 4 + 7 = 11, so (−3,11)(-3, 11) is on the line.
3. xx-intercept: set y=0  ⟹  −43x+7=0  ⟹  x=214=5.25y = 0 \implies -\frac{4}{3}x + 7 = 0 \implies x = \frac{21}{4} = 5.25, so (7,0)(7,0) is incorrect.
4. Quadrants: A line with negative slope and positive yy-intercept covers Quadrants I, II, and IV only, so passing through Quadrant III is false.
5. Intersection with line KK: set 34x−3=−43x+7  ⟹  2512x=10  ⟹  x=4.8\frac{3}{4}x - 3 = -\frac{4}{3}x + 7 \implies \frac{25}{12}x = 10 \implies x = 4.8, y=0.6y = 0.6, which is in Quadrant I, not Quadrant IV.
Direct algebraic verification confirms which geometric properties hold true for line MM.

Key Concept

Perpendicular Slopes and Linear Properties in Coordinate Geometry
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Topics

Coordinate Geometry and Lines

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