Examkin

Question

Difficulty: HardLinear Functions and Graphs

In the xyxy-plane, the graph of a linear function ff has a yy-intercept of (0,r)(0, r) and an xx-intercept of (s,0)(s, 0), where rr and ss are nonzero constants. A second line, gg, is perpendicular to the graph of ff and passes through the point (s,r)(s, r). Which of the following expressions represents the xx-intercept of line gg?

  1. A
    s2−r2r\frac{s^2 - r^2}{r}
  2. B
    s2+r2s\frac{s^2 + r^2}{s}
  3. s2−r2s\frac{s^2 - r^2}{s}Answer
  4. D
    r2−s2r\frac{r^2 - s^2}{r}

Answer

The expression s2−r2s\frac{s^2 - r^2}{s}
The slope of line ff is found using its intercepts (0,r)(0, r) and (s,0)(s, 0), which gives a slope of mf=−rsm_f = -\frac{r}{s}. Since line gg is perpendicular to line ff, its slope is the negative reciprocal, mg=srm_g = \frac{s}{r}. Using the point-slope form with the point (s,r)(s, r), the equation for line gg is y−r=sr(x−s)y - r = \frac{s}{r}(x - s). Setting y=0y = 0 to find the xx-intercept yields x=s2−r2sx = \frac{s^2 - r^2}{s}.

Step-by-Step Solution

1
Determine the slope of the linear function ff using its intercepts.
The slope of ff is mf=0−rs−0=−rsm_f = \frac{0 - r}{s - 0} = -\frac{r}{s}.
The slope of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Find the slope of line gg, which is perpendicular to the graph of ff.
The slope of line gg is mg=−1mf=−1−r/s=srm_g = -\frac{1}{m_f} = -\frac{1}{-r/s} = \frac{s}{r}.
Perpendicular lines have slopes that are negative reciprocals of each other.
3
Write the equation of line gg in point-slope form using the point (s,r)(s, r).
The equation of line gg is y−r=sr(x−s)y - r = \frac{s}{r}(x - s).
The point-slope form of a linear equation is y−y1=m(x−x1)y - y_1 = m(x - x_1).
4
Solve for the xx-intercept of line gg by setting y=0y = 0.
Setting y=0y = 0 gives −r=sr(x−s)  ⟹  −r2s=x−s  ⟹  x=s−r2s=s2−r2s-r = \frac{s}{r}(x - s) \implies -\frac{r^2}{s} = x - s \implies x = s - \frac{r^2}{s} = \frac{s^2 - r^2}{s}.
The xx-intercept of a line is the value of xx when y=0y = 0.

Key Concept

Finding the equation of a perpendicular line and its intercepts using algebraic representations.
Rate this question

Topics

Linear Functions and Graphs

More questions from this topic