In the xy-plane, line k passes through the points (-3, 5) and (1, -3). Line m is perpendic… — GRE General Test

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(1, -3)$ . Line $m$ is perpendicular to line $k$ at line $k$ 's $x$ -intercept. What is the $y$ -intercept of line $m$ ?

$\frac{1}{4}$ Answer
B
$-\frac{1}{4}$
C
$10$
D
$-1$
E
$-4$

Answer

The

y

-intercept of line

m

\frac{1}{4}

The line

k

has a slope of

-2

and an

x

-intercept of

\left(-\frac{1}{2}, 0\right)

. A line perpendicular to line

k

must have a slope of

\frac{1}{2}

. Substituting the point

\left(-\frac{1}{2}, 0\right)

into the line equation yields

y = \frac{1}{2}x + \frac{1}{4}

, so the

y

-intercept is

\frac{1}{4}

Step-by-Step Solution

Calculate the slope of line

k

Slope

m_k = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

Find the equation of line

k

and determine its

x

-intercept

Line

k

equation:

y - (-3) = -2(x - 1) \implies y = -2x - 1

. Setting

y = 0

gives

0 = -2x - 1 \implies x = -\frac{1}{2}

. The

x

-intercept is

\left(-\frac{1}{2}, 0\right)

The

x

-intercept is the point where the line crosses the

x

-axis (

y = 0

Find the slope of line

m

Slope

m_m = -\frac{1}{m_k} = -\frac{1}{-2} = \frac{1}{2}

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

Determine the equation of line

m

and its

y

-intercept

Using point-slope form with

\left(-\frac{1}{2}, 0\right)

and slope

\frac{1}{2}

y - 0 = \frac{1}{2}\left(x - \left(-\frac{1}{2}\right)\right) \implies y = \frac{1}{2}x + \frac{1}{4}

. Setting

x = 0

gives

y = \frac{1}{4}

The

y

-intercept is the constant term

b

when written in slope-intercept form

y = mx + b

Key Concept

Perpendicular line slopes and intercept calculations

Estimated Time:2m 0s

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