In the xy-plane, line L_1 passes through the origin (0,0) and the point (4, 3). Line L_2 i… — GRE General Test

In the $xy$ -plane, line $L_1$ passes through the origin $(0,0)$ and the point $(4, 3)$ . Line $L_2$ is formed by reflecting line $L_1$ across the vertical line $x = 2$ and then translating the resulting line downward by $5$ units. If line $L_2$ intersects the $y$ -axis at the point $(0, k)$ , what is the value of $k$ ?

A
$-5$
$-2$ Answer
C
$\frac{1}{3}$
D
$3$
E
$8$

Answer

The correct value of

k

-2

Line

L_1

has slope

\frac{3}{4}

and equation

y = \frac{3}{4}x

. Reflecting across

x = 2

replaces

x

with

4 - x

, transforming the equation into

y = \frac{3}{4}(4 - x) = 3 - \frac{3}{4}x

. Translating downward by

5

units gives

y = -\frac{3}{4}x - 2

. Setting

x = 0

gives the

y

-intercept

(0, -2)

, making

-2

the correct value.

Step-by-Step Solution

Find the equation of line

L_1

The slope of line

L_1

passing through

(0,0)

and

(4,3)

m = \frac{3 - 0}{4 - 0} = \frac{3}{4}

. Thus, the equation is

y = \frac{3}{4}x

Establishing the initial linear equation is necessary before applying coordinate transformations.

Apply the reflection across the line

x = 2

Reflecting any point

(x, y)

across

x = 2

transforms its x-coordinate to

2(2) - x = 4 - x

. Substituting

4 - x

into the equation gives

y = \frac{3}{4}(4 - x) = 3 - \frac{3}{4}x

Reflection across a vertical line

x = a

preserves the y-values while mapping

x \mapsto 2a - x

Apply the downward vertical translation by

5

units.

Subtracting

5

from the equation yields

y = \left(3 - \frac{3}{4}x\right) - 5 = -\frac{3}{4}x - 2

Translating a graph downward by

c

units subtracts

c

from the output

y

Determine the y-intercept of line

L_2

Setting

x = 0

y = -\frac{3}{4}x - 2

yields

y = -2

, so

k = -2

The y-intercept of a line occurs where

x = 0

Key Concept

Coordinate Transformations of Lines: Reflection across

x = a

and Vertical Translations

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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