In the xy-plane, a line with a negative slope passes through the point (4, 3) and intersec… — SAT

In the $xy$ -plane, a line with a negative slope passes through the point $(4, 3)$ and intersects the positive $x$ -axis at $(a, 0)$ and the positive $y$ -axis at $(0, b)$ . If the area of the triangle formed by this line and the coordinate axes is $24$ , what is the value of $b$ ?

Answer: 6

Answer

The value of

b

is 6.

The correct value is 6. By representing the linear equation in intercept form as

\frac{x}{a} + \frac{y}{b} = 1

and substituting the given point

(4, 3)

, we obtain

\frac{4}{a} + \frac{3}{b} = 1

. Using the area of the triangle,

\frac{1}{2}ab = 24

, we can write

a = \frac{48}{b}

. Substituting this into the intercept equation yields

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

leads to the quadratic equation

b^2 - 12b + 36 = 0

, which factors as

(b-6)^2 = 0

, giving the unique solution

b = 6

Step-by-Step Solution

Express the line in intercept form

\frac{x}{a} + \frac{y}{b} = 1

Since the intercepts are

(a, 0)

and

(0, b)

with

a > 0

and

b > 0

, the intercept form of a linear equation is the most direct representation.

Substitute the given point

(4, 3)

into the intercept form

\frac{4}{a} + \frac{3}{b} = 1

The line passes through

(4, 3)

, so these coordinates must satisfy the equation of the line.

Express

a

in terms of

b

using the area of the triangle

a = \frac{48}{b}

The area of the right triangle with base

a

and height

b

\frac{1}{2}ab = 24

, which gives

ab = 48

Substitute

a = \frac{48}{b}

into the equation from Step 2

\frac{b}{12} + \frac{3}{b} = 1

This reduces the equation to a single variable

b

Clear denominators and write the equation in standard quadratic form

b^2 - 12b + 36 = 0

Multiplying both sides by

12b

allows us to form a quadratic equation.

Factor the quadratic equation to solve for

b

b = 6

The quadratic expression is a perfect square trinomial,

(b-6)^2 = 0

, which yields

b = 6

Key Concept

Using the intercept form of a linear equation and geometric properties of linear graphs to solve for unknown parameters.

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