In the standard (x, y) coordinate plane, a line L passes through the point (2, -3) and doe… — ACT

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, -3)$ and does not pass through the origin. The distance from the origin to the line's $x$ -intercept is twice the distance from the origin to the line's $y$ -intercept. Which of the following could be the slope of line $L$ ?

A
$-2$
$-\frac{1}{2}$ Answer
C
$\frac{2}{3}$
D
$1$
E
$2$

Answer

The slope of the line could be

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. By setting the

x

-intercept of the line to

(a, 0)

and the

y

-intercept to

(0, b)

, the distance condition gives

|a| = 2|b|

. Substituting the point

(2, -3)

into the intercept equation of a line,

\frac{x}{a} + \frac{y}{b} = 1

, yields

\frac{2}{a} - \frac{3}{b} = 1

. Solving the two cases (

a = 2b

and

a = -2b

) gives the intercept pairs

(-4, -2)

and

(8, -4)

. Calculating the slope

m = -\frac{b}{a}

for both pairs results in

-\frac{1}{2}

and

\frac{1}{2}

. Since

-\frac{1}{2}

is one of these possible values, it is the correct choice.

Step-by-Step Solution

Represent the line

L

using its intercepts. Let the

x

-intercept of

L

(a, 0)

and the

y

-intercept be

(0, b)

. Since the line does not pass through the origin, we have

a \neq 0

and

b \neq 0

The equation of the line can be written in intercept form as:

\frac{x}{a} + \frac{y}{b} = 1

Using intercept form allows us to directly relate the coordinates of the intercepts to the given point and the distance condition.

Translate the distance condition and the given point into equations. The distance from the origin to the

x

-intercept is

|a|

, and the distance to the

y

-intercept is

|b|

. The problem states that the distance to the

x

-intercept is twice the distance to the

y

-intercept, so

|a| = 2|b|

. Additionally, the line passes through

(2, -3)

, so we substitute

x = 2

and

y = -3

into the intercept equation.

|a| = 2|b|

and

\frac{2}{a} - \frac{3}{b} = 1

This sets up a system of equations to solve for the unknown intercepts

a

and

b

Solve the system of equations by analyzing the two cases for the absolute value:

a = 2b

and

a = -2b

Case 1: If

a = 2b

, then

\frac{2}{2b} - \frac{3}{b} = 1 \implies \frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{2}{b} = 1 \implies b = -2

, which gives

a = -4

. Case 2: If

a = -2b

, then

\frac{2}{-2b} - \frac{3}{b} = 1 \implies -\frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{4}{b} = 1 \implies b = -4

, which gives

a = 8

Resolving the absolute value yields the exact coordinates of the intercepts for both valid scenarios.

Calculate the slope

m

for both cases using the formula

m = -\frac{b}{a}

For Case 1,

m = -\frac{-2}{-4} = -\frac{1}{2}

. For Case 2,

m = -\frac{-4}{8} = \frac{1}{2}

The slope of a line with intercepts

(a, 0)

and

(0, b)

is given by

-\frac{b}{a}

Key Concept

Slope of a Line

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