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

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(3, 1)$ . Line $m$ is perpendicular to line $k$ and passes through the point $(1, -1)$ . Which of the following statements must be true? Indicate all such statements.

The slope of line $m$ is $\frac{3}{2}$ .Cevap
Line $m$ passes through Quadrants I, III, and IV.Cevap
The $x$ -intercept of line $k$ is $\left(\frac{9}{2}, 0\right)$ .Cevap
D
The slope of line $k$ is $-\frac{3}{2}$ .
E
Line $k$ and line $m$ intersect at the point $(3, 1)$ .

Cevap

The statements confirming that the slope of line

m

\frac{3}{2}

, line

m

passes through Quadrants I, III, and IV, and the

x

-intercept of line

k

\left(\frac{9}{2}, 0\right)

are all correct.

The slope of line

k

-\frac{2}{3}

, making line

m

's perpendicular slope

\frac{3}{2}

. The equation for line

m

y = \frac{3}{2}x - \frac{5}{2}

, which crosses the

y

-axis at

\left(0, -\frac{5}{2}\right)

and the

x

-axis at

\left(\frac{5}{3}, 0\right)

, thus entering Quadrants I, III, and IV. The

x

-intercept of line

k

is found by setting

y = 0

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

, yielding

x = \frac{9}{2}

Adım Adım Çözüm

Calculate the slope of line

k

using the slope formula

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

Slope of line

k

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

Determining the slope of line

k

is required to find perpendicular slopes and line equations.

Determine the slope and equation of line

m

Perpendicular slope is

m_m = -\frac{1}{m_k} = \frac{3}{2}

. Using point-slope form with point

(1, -1)

y - (-1) = \frac{3}{2}(x - 1) \Rightarrow y = \frac{3}{2}x - \frac{5}{2}

Perpendicular lines have negative reciprocal slopes.

Find the

x

-intercept of line

k

and analyze the quadrant passage of line

m

Line

k

equation is

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

; setting

y = 0

yields

x = \frac{9}{2}

. For line

m

, positive slope and negative

y

-intercept

\left(0, -\frac{5}{2}\right)

mean it traverses Quadrants I, III, and IV.

Setting

y=0

gives the

x

-intercept, and evaluating the line equation across negative, zero, and positive

x

-values identifies quadrant coverage.

Anahtar Kavram

Coordinate Geometry: Line equations, slopes of perpendicular lines, intercepts, and quadrant passage.

Soru

Cevap

Adım Adım Çözüm

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