Finding Line Equations Using Axis Intercepts

This problem requires us to find the equations of two lines based on the given sum and product of the intercepts they form on the coordinate axes.

Understanding the Intercept Form

The equation of a line can be represented in the intercept form as:

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

where '$a$' is the x-intercept (the point where the line crosses the x-axis) and '$b$' is the y-intercept (the point where the line crosses the y-axis).

Applying the Given Conditions

We are given the following information about the intercepts '$a$' and '$b$':

• Product of intercepts: $a \times b = -8$
• Sum of intercepts: $a + b = 2$

Solving for the Intercepts

We need to find the values of '$a$' and '$b$' that satisfy both conditions. We have a system of two equations:

1. $$ ab = -8 $$
2. $$ a + b = 2 $$

From the second equation, we can express '$b$' in terms of '$a$':

$$ b = 2 - a $$

Substitute this expression for '$b$' into the first equation:

$$ a(2 - a) = -8 $$

Distribute '$a$':

$$ 2a - a^2 = -8 $$

Rearrange the equation into the standard quadratic form ($ax^2 + bx + c = 0$):

$$ a^2 - 2a - 8 = 0 $$

Factor the quadratic equation. We look for two numbers that multiply to -8 and add up to -2. These numbers are -4 and +2.

$$ (a - 4)(a + 2) = 0 $$

This gives two possible values for '$a$':

• $a - 4 = 0 \implies a = 4$
• $a + 2 = 0 \implies a = -2$

Determining the Line Equations

Now, we find the corresponding '$b$' value for each '$a$' value and write the line equation.

Scenario 1: $a = 4$

Using $b = 2 - a$, we get:

$$ b = 2 - 4 = -2 $$

Substitute $a=4$ and $b=-2$ into the intercept form:

$$ \frac{x}{4} + \frac{y}{-2} = 1 $$

To clear the fractions, multiply the entire equation by the least common multiple of 4 and -2, which is 4:

$$ 4 \left( \frac{x}{4} \right) + 4 \left( \frac{y}{-2} \right) = 4 \times 1 $$

$$ x - 2y = 4 $$

Scenario 2: $a = -2$

Using $b = 2 - a$, we get:

$$ b = 2 - (-2) = 2 + 2 = 4 $$

Substitute $a=-2$ and $b=4$ into the intercept form:

$$ \frac{x}{-2} + \frac{y}{4} = 1 $$

To clear the fractions, multiply the entire equation by the least common multiple of -2 and 4, which is 4:

$$ 4 \left( \frac{x}{-2} \right) + 4 \left( \frac{y}{4} \right) = 4 \times 1 $$

$$ -2x + y = 4 $$

Final Equations

The two equations of the lines satisfying the given conditions are:

These correspond to the equations found in the correct answer option.