In the xy-plane, line l passes through the point (3, 4) and has a slope of m, where m > 0.… — SAT

In the $xy$ -plane, line $l$ passes through the point $(3, 4)$ and has a slope of $m$ , where $m > 0$ . Line $k$ is perpendicular to line $l$ and passes through the point $(2, 2)$ . If the sum of the $y$ -intercepts of line $l$ and line $k$ is $1$ , what is the value of $m$ ?

A
$\frac{1}{3}$
B
$\frac{2}{3}$
C
$1$
$2$ Answer

Answer

The correct value of

m

2

The correct answer is the value

2

. By finding the equations of both lines in slope-intercept form, we express their

y

-intercepts in terms of

m

: the

y

-intercept of line

l

4 - 3m

and the

y

-intercept of line

k

\frac{2}{m} + 2

. Setting their sum to

1

gives the equation

3m^2 - 5m - 2 = 0

. Factoring this quadratic yields the solutions

m = -\frac{1}{3}

and

m = 2

. Since

m

must be positive,

m = 2

is the only valid solution.

Step-by-Step Solution

Find the equation and

y

-intercept of line

l

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 4)

and slope

m

, the equation of line

l

y - 4 = m(x - 3)

, which simplifies to

y = mx + 4 - 3m

. Thus, the

y

-intercept of line

l

4 - 3m

Expressing the

y

-intercept of line

l

in terms of

m

allows us to use it in the sum equation.

Find the equation and

y

-intercept of line

k

Since line

k

is perpendicular to line

l

, its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Using the point-slope form with the point

(2, 2)

, the equation of line

k

y - 2 = -\frac{1}{m}(x - 2)

, which simplifies to

y = -\frac{1}{m}x + \frac{2}{m} + 2

. Thus, the

y

-intercept of line

k

\frac{2}{m} + 2

Expressing the

y

-intercept of line

k

in terms of

m

allows us to use it in the sum equation.

Set up the equation for the sum of the

y

-intercepts and solve for

m

The sum of the

y

-intercepts is

(4 - 3m) + (\frac{2}{m} + 2) = 1

. Simplifying this equation gives

6 - 3m + \frac{2}{m} = 1 \implies 5 - 3m + \frac{2}{m} = 0

. Multiplying by

m

yields

3m^2 - 5m - 2 = 0

. Factoring the quadratic gives

(3m + 1)(m - 2) = 0

, which has solutions

m = -\frac{1}{3}

and

m = 2

. Since

m > 0

, we have

m = 2

To determine the unique positive slope that satisfies the given conditions.

Key Concept

Writing equations of perpendicular lines and finding their intercepts using slopes and points.

Question

Answer

Step-by-Step Solution

Key Concept

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