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Zorluk: OrtaParallel and Perpendicular Lines

A line, L1L_1, contains the points (2,5)(2, 5) and (6,−3)(6, -3) in a coordinate plane. Another line, L2L_2, is perpendicular to L1L_1 and is defined by the equation ax+2y=7ax + 2y = 7. What is the value of aa?

  1. A
    -4
  2. -1Cevap
  3. C
    1
  4. D
    2
  5. E
    4

Cevap

The value of aa is −1-1.
The slope of line L1L_1 is −2-2. A line perpendicular to it must have a slope that is the negative reciprocal, which is 12\frac{1}{2}. Rewriting ax+2y=7ax + 2y = 7 in slope-intercept form gives y=−a2x+72y = -\frac{a}{2}x + \frac{7}{2}, where the slope is −a2-\frac{a}{2}. Setting this slope equal to 12\frac{1}{2} gives −a2=12-\frac{a}{2} = \frac{1}{2}, which simplifies to a=−1a = -1.

Adım Adım Çözüm

1
Find the slope of line L1L_1 using the points (2,5)(2, 5) and (6,−3)(6, -3).
m1=−3−56−2=−84=−2m_1 = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2.
The slope formula is the change in y divided by the change in x.
2
Determine the perpendicular slope for L2L_2.
m2=−1m1=12m_2 = -\frac{1}{m_1} = \frac{1}{2}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Rewrite the equation of L2L_2, ax+2y=7ax + 2y = 7, in slope-intercept form to find its slope expression.
2y=−ax+7⇒y=−a2x+722y = -ax + 7 \Rightarrow y = -\frac{a}{2}x + \frac{7}{2}. The slope is −a2-\frac{a}{2}.
Slope-intercept form y=mx+by = mx + b allows direct identification of the slope coefficient.
4
Equate the slope of L2L_2 to the perpendicular slope and solve for aa.
−a2=12⇒a=−1-\frac{a}{2} = \frac{1}{2} \Rightarrow a = -1.
Solving the equation gives the value of aa required for the lines to be perpendicular.

Anahtar Kavram

Perpendicular lines have slopes that are negative reciprocals of each other.
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Konular

Parallel and Perpendicular Lines

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