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Zorluk: ZorSlope of a Line

A line in the standard (x,y)(x, y) coordinate plane has a slope of 34\frac{3}{4} and contains the points P(a,3)P(a, 3) and Q(5,b)Q(5, b). If the distance between PP and QQ is 1010 units, what is one possible value of a+ba + b?

  1. A
    −6-6
  2. B
    22
  3. 66Cevap
  4. D
    1212
  5. E
    2222

Cevap

The correct answer is 66.
The correct answer is 66. Using the slope definition m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}, we write the slope equation as b−35−a=34\frac{b - 3}{5 - a} = \frac{3}{4}, which gives b−3=34(5−a)b - 3 = \frac{3}{4}(5 - a). Substituting this expression into the distance formula d=(5−a)2+(b−3)2=10d = \sqrt{(5 - a)^2 + (b - 3)^2} = 10 yields 10=(5−a)2+(34(5−a))2=54∣5−a∣10 = \sqrt{(5 - a)^2 + \left(\frac{3}{4}(5 - a)\right)^2} = \frac{5}{4}|5 - a|. Solving for the horizontal change gives ∣5−a∣=8|5 - a| = 8, which means 5−a=±85 - a = \pm 8. If 5−a=85 - a = 8, then a=−3a = -3 and b−3=6⇒b=9b - 3 = 6 \Rightarrow b = 9. The sum of these coordinates is −3+9=6-3 + 9 = 6.

Adım Adım Çözüm

1
Set up the slope equation using the definition of slope: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
b−35−a=34\frac{b - 3}{5 - a} = \frac{3}{4}, which simplifies to b−3=34(5−a)b - 3 = \frac{3}{4}(5 - a).
This establishes a relationship between the coordinate differences of points PP and QQ based on the given slope.
2
Set up the distance equation using the distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
10=(5−a)2+(b−3)210 = \sqrt{(5 - a)^2 + (b - 3)^2}.
This uses the given distance of 1010 units to constrain the coordinate differences.
3
Substitute the slope relationship b−3=34(5−a)b - 3 = \frac{3}{4}(5 - a) into the distance equation and solve for (5−a)(5 - a).
10=(5−a)2+(34(5−a))2⇒10=(5−a)2(1+916)⇒10=54∣5−a∣⇒∣5−a∣=810 = \sqrt{(5 - a)^2 + \left(\frac{3}{4}(5 - a)\right)^2} \Rightarrow 10 = \sqrt{(5 - a)^2 \left(1 + \frac{9}{16}\right)} \Rightarrow 10 = \frac{5}{4}|5 - a| \Rightarrow |5 - a| = 8. Thus, 5−a=85 - a = 8 or 5−a=−85 - a = -8.
Substituting one variable simplifies the system to a single quadratic equation in terms of the horizontal change.
4
Solve for the two possible values of aa and calculate the corresponding values of bb.
Case 1: If 5−a=85 - a = 8, then a=−3a = -3. Substituting this back gives b−3=34(8)=6⇒b=9b - 3 = \frac{3}{4}(8) = 6 \Rightarrow b = 9. The sum is a+b=−3+9=6a + b = -3 + 9 = 6. Case 2: If 5−a=−85 - a = -8, then a=13a = 13. Substituting this back gives b−3=34(−8)=−6⇒b=−3b - 3 = \frac{3}{4}(-8) = -6 \Rightarrow b = -3. The sum is a+b=13+(−3)=10a + b = 13 + (-3) = 10.
Evaluating both branches of the absolute value equation yields the two valid coordinate combinations that satisfy both the slope and distance criteria.

Anahtar Kavram

Calculating the slope of a line and using it in conjunction with the distance formula to find unknown coordinates on the coordinate plane.
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Konular

Slope of a Line

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