In the xy-plane, a line with a positive slope m and a y-intercept of (0, b) passes through… — SAT

In the $xy$ -plane, a line with a positive slope $m$ and a $y$ -intercept of $(0, b)$ passes through the point $(b, 12)$ , where $b > 0$ . If the area of the triangle bounded by the line, the $x$ -axis, and the $y$ -axis is $4$ , what is the value of $m$ ?

Answer: 2

Answer

The line equation is

y = mx + b

. Substituting the point

(b, 12)

gives

12 = b(m + 1)

. The

x

-intercept of the line is at

(-\frac{b}{m}, 0)

, which gives a base length of

\frac{b}{m}

for the right triangle, while the height is

b

. The area of the triangle is

\frac{b^2}{2m} = 4 \implies b^2 = 8m

. Substituting

b = \frac{12}{m + 1}

into this equation yields

\frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Solving this for positive values of

m

gives the unique solution

m = 2

Step-by-Step Solution

Express the line equation and substitute the given point

(b, 12)

b(m + 1) = 12

Since the line has a

y

-intercept of

(0, b)

and a slope of

m

, its equation is

y = mx + b

. Substituting the point

(b, 12)

yields

12 = m(b) + b = b(m + 1)

Determine the lengths of the base and height of the triangle formed by the line and the coordinate axes.

Base

= \frac{b}{m}

, Height

= b

The

y

-intercept is at

(0, b)

and the

x

-intercept is at

(-\frac{b}{m}, 0)

. Since

b > 0

and

m > 0

, the height is

b

and the base is the absolute value of the

x

-intercept, which is

\frac{b}{m}

Write the area equation of the triangle and simplify it.

b^2 = 8m

The area of the right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \left(\frac{b}{m}\right)b = \frac{b^2}{2m}

. Setting this equal to the given area of

4

gives

\frac{b^2}{2m} = 4 \implies b^2 = 8m

Solve the system of equations for

m

m = 2

Substituting

b = \frac{12}{m + 1}

into

b^2 = 8m

yields

\left(\frac{12}{m + 1}\right)^2 = 8m \implies \frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Since

m(m + 1)^2

is strictly increasing for positive

m

, testing small integers shows

2(3)^2 = 18

, so

m = 2

is the unique positive real solution.

Key Concept

Formulating linear equations in slope-intercept form and solving non-linear systems of equations derived from geometric constraints.

Alternative Method

Instead of algebraically expanding the cubic equation, one can test small positive integers for

m

in the equation

m(m + 1)^2 = 18

. Testing

m = 1

yields

1(2)^2 = 4

(too small), and testing

m = 2

yields

2(3)^2 = 18

(correct). Because

m(m + 1)^2

is strictly increasing for

m > 0

m = 2

is the only positive real root.

Estimated Time:3m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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