In the xy-plane, line k has an x-intercept of (8, 0) and a y-intercept of (0, 6). Line p i… — GRE General Test

In the $xy$ -plane, line $k$ has an $x$ -intercept of $(8, 0)$ and a $y$ -intercept of $(0, 6)$ . Line $p$ is perpendicular to line $k$ and intersects line $k$ at its $y$ -intercept. What is the $x$ -intercept of line $p$ ?

A
$-8$
$-\frac{9}{2}$ Answer
C
$\frac{9}{2}$
D
$8$
E
$-\frac{32}{3}$

Answer

-\frac{9}{2}

The line

k

passes through

(8, 0)

and

(0, 6)

, giving a slope of

m_k = \frac{6 - 0}{0 - 8} = -\frac{3}{4}

. A line perpendicular to

k

must have a slope equal to the negative reciprocal of

-\frac{3}{4}

, which is

\frac{4}{3}

. Because line

p

intersects line

k

(0, 6)

, its

y

-intercept is also

6

, making its equation

y = \frac{4}{3}x + 6

. Setting

y = 0

gives

0 = \frac{4}{3}x + 6

, which solves to

x = -\frac{9}{2}

Step-by-Step Solution

Calculate the slope of line

k

Slope of line

k

m_k = \frac{6 - 0}{0 - 8} = -\frac{6}{8} = -\frac{3}{4}

The slope formula is

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

using given intercepts

(8, 0)

and

(0, 6)

Determine the slope of line

p

Slope of line

p

m_p = -\frac{1}{m_k} = \frac{4}{3}

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Write the equation of line

p

Equation of line

p

y = \frac{4}{3}x + 6

Line

p

intersects line

k

at its

y

-intercept

(0, 6)

, so line

p

has a

y

-intercept of

6

Find the

x

-intercept of line

p

x = -\frac{9}{2}

(or

-4.5

)

Set

y = 0

in the line equation:

0 = \frac{4}{3}x + 6 \implies \frac{4}{3}x = -6 \implies x = -6 \cdot \frac{3}{4} = -\frac{9}{2}

Key Concept

Perpendicular Slopes and Line Intercepts

Estimated Time:1m 30s

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