In the standard (x, y) coordinate plane, if the three distinct points (k, 4), (4, k), and … — ACT

In the standard $(x, y)$ coordinate plane, if the three distinct points $(k, 4)$ , $(4, k)$ , and $(5, 6)$ are collinear, then there are exactly two possible values for the slope of the line passing through these points.

Answer: Answer

Answer

The statement is false because the requirement that the three points must be distinct eliminates one of the algebraic solutions for

k

, resulting in only one possible slope for the line.

The statement is false because equating the slopes between the points leads to the quadratic equation

k^2 - 11k + 28 = 0

, which has the solutions

k = 4

and

k = 7

. However, substituting

k = 4

back into the coordinates results in the points

(4, 4)

and

(4, 4)

, which are not distinct. Thus, only

k = 7

is a valid solution, which yields exactly one line with a slope of

-1

Step-by-Step Solution

Set up the collinearity condition using the slope formula.

The slope of the segment connecting

(4, k)

and

(5, 6)

m_1 = \frac{6 - k}{5 - 4} = 6 - k

. The slope of the segment connecting

(k, 4)

and

(5, 6)

m_2 = \frac{6 - 4}{5 - k} = \frac{2}{5 - k}

. Equating the two slopes gives

6 - k = \frac{2}{5 - k}

For three points to be collinear, the slope between any two pairs of points must be equal.

Solve the rational equation for

k

Multiplying both sides by

5 - k

yields

(6 - k)(5 - k) = 2

, which expands to

30 - 11k + k^2 = 2

. Rearranging into standard quadratic form gives

k^2 - 11k + 28 = 0

. Factoring the quadratic yields

(k - 4)(k - 7) = 0

, giving the solutions

k = 4

k = 7

Finding the roots of the quadratic equation identifies the algebraic candidates for

k

Verify if the points are distinct for each candidate value of

k

k = 4

, the first two points are

(4, 4)

and

(4, 4)

, which are not distinct. If

k = 7

, the three points are

(7, 4)

(4, 7)

, and

(5, 6)

, which are distinct.

The problem specifies that the three points must be distinct, so any solution violating this condition must be discarded.

Determine the number of possible slopes.

For the only valid value

k = 7

, the slope of the line is

m = 6 - 7 = -1

. There is only one possible slope.

Comparing the number of valid slopes to the statement shows that the statement is false.

Key Concept

Collinearity of points in the coordinate plane and constraints on slope calculation.

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