A line with a slope of -2 passes through the point (1, 7) and contains the point P(a, 3). … — GRE General Test

A line with a slope of $-2$ passes through the point $(1, 7)$ and contains the point $P(a, 3)$ . What is the distance between point $P$ and the point $Q(11, -3)$ ?

A
$2\sqrt{10}$
B
$6\sqrt{5}$
$10$ Answer
D
$100$
E
$14$

Answer

The distance between point

P

and point

Q

10

Using the slope definition

m = \frac{\Delta y}{\Delta x}

, we set up

\frac{3 - 7}{a - 1} = -2

, which solves to

a = 3

, giving point

P(3, 3)

. Applying the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

between

P(3, 3)

and

Q(11, -3)

yields

\sqrt{(11 - 3)^2 + (-3 - 3)^2} = \sqrt{64 + 36} = 10

Step-by-Step Solution

Use the slope formula to find the missing coordinate

a

of point

P(a, 3)

The slope

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{a - 1} = -2

, which simplifies to

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

, yielding

a - 1 = 2

, so

a = 3

The slope between any two points on a straight line must equal the given slope of

-2

Identify the coordinates of point

P

Point

P

has coordinates

(3, 3)

Substituting

a = 3

into

P(a, 3)

gives the exact position of

P

Apply the distance formula between

P(3, 3)

and

Q(11, -3)

d = \sqrt{(11 - 3)^2 + (-3 - 3)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10

The distance formula calculates the Euclidean distance between two points in the coordinate plane.

Key Concept

Line slope equation and coordinate distance formula

Alternative Method

Find the line equation directly in slope-intercept form:

y - 7 = -2(x - 1) \implies y = -2x + 9

. Substitute

y = 3

to get

3 = -2a + 9 \implies a = 3

. Then compute the distance between

(3, 3)

and

(11, -3)

using the standard distance formula.

Estimated Time:1m 30s

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