Line Q passes through the coordinates (-1, 7) and (7, 1) in the Cartesian plane. Line P is… — GRE General Test

Line $Q$ passes through the coordinates $(-1, 7)$ and $(7, 1)$ in the Cartesian plane. Line $P$ is perpendicular to line $Q$ and intersects the $x$ -axis at $(-6, 0)$ . What is the $y$ -intercept of line $P$ ?

Answer: 8

Answer

The

y

-intercept of line

P

is 8.

The slope of line

Q

is calculated as

\frac{1 - 7}{7 - (-1)} = -\frac{3}{4}

. Since line

P

is perpendicular to line

Q

, its slope is the negative reciprocal,

\frac{4}{3}

. Using the

x

-intercept point

(-6, 0)

, the line equation is

y - 0 = \frac{4}{3}(x + 6)

, which simplifies to

y = \frac{4}{3}x + 8

. Setting

x = 0

yields the

y

-intercept of

8

Step-by-Step Solution

Calculate the slope of line

Q

Slope of line

Q

-\frac{3}{4}

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 7}{7 - (-1)} = \frac{-6}{8} = -\frac{3}{4}

Determine the slope of perpendicular line

P

Slope of line

P

\frac{4}{3}

Perpendicular lines have slopes that are negative reciprocals of each other:

m_P = -\frac{1}{m_Q} = -\frac{1}{-3/4} = \frac{4}{3}

Find the equation and

y

-intercept of line

P

The

y

-intercept is

8

Line

P

passes through

(-6, 0)

. Using point-slope form:

y - 0 = \frac{4}{3}(x - (-6)) \implies y = \frac{4}{3}x + 8

. Substituting

x = 0

gives

y = 8

Key Concept

Perpendicular lines and line equations in coordinate geometry

Question

Answer

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