In the xy-plane, line m is defined by the equation ax + by = c, where a, b, and c are non-zero real numbers such that ab 0. Line k is perpendicular to line m and intersects line m at its x-intercept. Which of the following statements must be true? Select all such statements.

Line $m$ has a positive slope.; Line $k$ passes through Quadrant II.

In the xy-plane, line m is defined by the equation ax + by = c, where a, b, and c are non-… — GRE General Test

In the $xy$ -plane, line $m$ is defined by the equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero real numbers such that $ab < 0$ and $ac > 0$ . Line $k$ is perpendicular to line $m$ and intersects line $m$ at its $x$ -intercept. Which of the following statements must be true? Select all such statements.

Line $m$ has a positive slope.Answer
Line $k$ passes through Quadrant II.Answer
C
The $y$ -intercept of line $k$ is negative.
D
The point of intersection of line $m$ and line $k$ lies in Quadrant I.
E
Line $m$ passes through Quadrant II.

Answer

The statements asserting that line

m

has a positive slope and that line

k

passes through Quadrant II must be true.

The statement regarding line

m

having a positive slope is correct because

ab < 0

implies

a

and

b

have opposite signs, making

-\frac{a}{b} > 0

. The statement asserting line

k

passes through Quadrant II is correct because line

k

possesses a negative slope

\frac{b}{a} < 0

and a positive

y

-intercept

-\frac{bc}{a^2} > 0

, ensuring it enters Quadrant II.

Step-by-Step Solution

Determine the slope and intercepts of line

m

Line

m

y = -\frac{a}{b}x + \frac{c}{b}

. Slope is

-\frac{a}{b} > 0

because

ab < 0

x

-intercept is

\left(\frac{c}{a}, 0\right)

where

\frac{c}{a} > 0

because

ac > 0

y

-intercept is

\left(0, \frac{c}{b}\right)

where

\frac{c}{b} < 0

because

b

and

c

have opposite signs.

Converting standard line equations to slope-intercept form exposes the signs of slopes and intercepts based on coefficient products.

Determine the slope, equation, and properties of line

k

Since line

k

is perpendicular to line

m

, its slope is the negative reciprocal of

-\frac{a}{b}

, which is

\frac{b}{a} < 0

. Line

k

passes through

\left(\frac{c}{a}, 0\right)

, giving equation

y = \frac{b}{a}\left(x - \frac{c}{a}\right) = \frac{b}{a}x - \frac{bc}{a^2}

Perpendicular lines have slopes whose product is

-1

Analyze quadrant coverage for both lines and verify statements.

Line

m

has positive slope and negative

y

-intercept

\implies

passes through Quadrants I, III, IV. Line

k

has negative slope and positive

y

-intercept

-\frac{bc}{a^2} > 0 \implies

passes through Quadrants I, II, IV. Intersection is at

\left(\frac{c}{a}, 0\right)

on the positive

x

-axis.

Systematic sign analysis determines quadrant trajectory and exact axis locations.

Key Concept

Properties of lines, perpendicular slopes, and sign analysis of intercepts in coordinate geometry.

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