Question

If the lines \(\mathrm{x}+(\mathrm{a}-1) \mathrm{y}+1=0\) and \(2 \mathrm{x}+\mathrm{a}^{2} \mathrm{y}-1=0\) are perpendicular then \(\ldots \ldots\) (a) \(|\mathrm{a}|=2\) (b) \(0<\mathrm{a}<1\) (c) \(-1<\mathrm{a}<1\) (d) \(a=-1\)

Short answer

The correct answer is (d) \(a = -1\). Two lines are perpendicular when the product of their slopes is equal to -1. After finding the slopes of both lines and equating their product to -1, we obtained the equation \((a-1)a^2=2\). Analyzing the given options, we found that the equation is satisfied when \(a = -1\).

Step-by-step solution

Find the slope of the first line

The given first line equation is \(\mathrm{x}+(\mathrm{a}-1) \mathrm{y}+1=0\). To find the slope, we can rewrite this equation in the slope-intercept form, i.e., \(y = mx + b\). Solving for 'y' gives, \(y = -\frac{1}{a-1}x - \frac{1}{a-1}\), and the slope of the first line, m1, is \(-\frac{1}{a-1}\).

Find the slope of the second line

The given second line equation is \(2\mathrm{x}+\mathrm{a}^{2} \mathrm{y}-1=0\). Similar to the first line, we rewrite this equation in the slope-intercept form. Solving for 'y' gives, \(y = -\frac{2}{a^2}x + \frac{1}{a^2}\), and the slope of the second line, m2, is \(-\frac{2}{a^2}\).

Check if lines are perpendicular and find 'a'

Two lines are perpendicular when the product of their slopes is equal to -1. Thus, we have the equation \(m1 \cdot m2 = -1\). Plugging the slopes of the lines, we get \[\left(-\frac{1}{a-1}\right)\left(-\frac{2}{a^2}\right) = -1.\] Simplify this equation to find 'a': \[\frac{2}{(a-1)a^2} = 1 \Rightarrow 2 = (a-1)a^2.\] Now, we analyze the options given and check which one satisfies the obtained equation. (a) If \(|a|=2\), it implies \(a = 2\) or \(a = -2\). Plugging these values into the equation we have \(2 = (1)(4)\) or \(2=(-3)(4)\). These conditions are not satisfied. (b) If \(0 < a < 1\), the expression \((a-1)a^2\) becomes negative, making it impossible for the equation to hold. (c) If \(-1 < a < 1\), the only value for which the expression \((a-1)a^2 = 2\) is satisfied in this range is when \(a = -1\). (d) If \(a = -1\), the equation becomes \(2 = (0)(1)\), which satisfies the given equation. Thus, the correct answer is (d) \(a = -1\).

Key concepts

Slope of a Line

Understanding the slope of a linear equation is fundamental in coordinate geometry. It represents how steep a line is and the direction in which it inclines or declines. The slope, often denoted as 'm', can be calculated if we know two points on the line or by rearranging the equation of the line into the slope-intercept form.

The slope from two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \.\]
When we have the equation of a line, first we should aim to reorganize it into the slope-intercept form to identify the slope directly from the equation. A positive slope indicates the line rises from left to right, whereas a negative slope means the line falls from left to right. A slope of zero suggests a horizontal line, and an undefined slope (or one that could be construed as 'infinity') indicates a vertical line.

Slope-Intercept Form

The slope-intercept form of a line is perhaps the most intuitive and widely-used representation of a linear equation. It is expressed as \( y = mx + b \), where 'm' represents the slope of the line and 'b' denotes the y-intercept, which is the point where the line crosses the y-axis.

Converting a linear equation to this form involves solving the equation for y, such that y is isolated on one side of the equation. This can often require rearranging terms and simplifying the expression.

Example:

For the equation \( ax + by = c \), we can solve for y to find the slope-intercept form: \[ y = -\frac{a}{b}x + \frac{c}{b} \.\]
This works for all non-vertical linear equations, and once in this form, the graph of the line and other characteristics, like whether lines are parallel or perpendicular, become much clearer.

Product of Slopes for Perpendicular Lines

In coordinate geometry, there's a very important relationship between the slopes of two perpendicular lines. If two lines are perpendicular to each other, the product of their slopes is -1. This is a consequence of the geometric fact that perpendicular lines intersect at a right angle.

Mathematically, if the slope of one line is \( m1 \) and the slope of the other line is \( m2 \) and the lines are perpendicular, then: \[ m1 \cdot m2 = -1 \.\]
This rule is especially useful when determining if two lines are perpendicular by calculation rather than visual assessment of a graph. Also, in the context of problem-solving, as demonstrated in our exercise, this property is vital for determining unknowns that relate to the slopes of lines that are given to be perpendicular.