Solving for the Equation of a Line Forming an Isosceles Triangle with Area 8 Units

Let's explore the problem of determining the equation of a line in the first quadrant which forms an isosceles triangle with the coordinate axes and has an area of 8 units. This problem is a blend of algebra and geometry, integrating equations of lines with the properties of triangle areas.

Formulating the Equation

To solve this problem, we first define the line in the first quadrant that forms an isosceles triangle with the coordinate axes. Let the x-intercept be 'a' and the y-intercept be 'b'. The area of the triangle formed by these intercepts and the origin is given by:

Area (frac{1}{2} cdot a cdot b 8)

Rewriting this, we get:

(frac{1}{2} cdot a cdot b 8)

(a cdot b 16)

This means the product of the intercepts is 16. Hence, the equation of the line in the intercept form is:

(frac{x}{a} frac{y}{16/a} 1)

Using a Specific Point to Determine Intercepts

We know the line passes through the point (4, 1). Substituting these coordinates into the line equation, we get:

(frac{4}{a} frac{1}{16/a} 1)

Multiplying through by (a cdot 16) to clear the denominators, we obtain:

(64 a 16a)

(a^2 - 16a 64 0)

This is a quadratic equation which can be factored as:

((a - 8)^2 0)

(a 8)

Thus, (a 8), and substituting (a 8) back into the equation (a cdot b 16), we have:

(8 cdot b 16)

(b 2)

Hence, the equation of the line is:

(frac{x}{8} frac{y}{2} 1)

Simplifying this, we get:

(x 4y 8)

Alternative Approach: Isosceles Right Triangle

Another approach is to consider the line as the base of an isosceles right triangle. For this setup, the slope of the line is (-1) (since the triangle must be isosceles with the same height and base). The area of the triangle formed is:

Area (frac{1}{2} cdot text{base} cdot text{height} 8)

Hence, the product of the intercepts (base and height) is:

(frac{1}{2} cdot a cdot a 8)

(a^2 16)

(a 4)

So, the intercepts are 4 and 4. Using the intercept form of the line's equation, we have:

(frac{x}{4} frac{y}{4} 1)

Simplifying this, we get:

(x y 4)

And, from another perspective, the equation of the line can also be expressed as (y -x 4).