Question

If \((1 / a),(1 / b),(1 / c)\) are in arithmetic sequence, then the line \((\mathrm{x} / \mathrm{a})+(\mathrm{y} / \mathrm{b})+(1 / \mathrm{c})=0\) passes through the fixed point \(\ldots \ldots\) (a) \((-1,-2)\) (b) \((-1,2)\) (c) \([1,-(1 / 2)]\) (d) \((1,-2)\)

Short answer

The fixed point through which the line passes is \(\boxed{\text{(c) } \left(1, -\frac{1}{2}\right)}\).

Step-by-step solution

Find the common difference of the arithmetic sequence

Since \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\) are in an arithmetic sequence, there must be a common difference between the terms. Let's call this common difference \(d\): \(\frac{1}{b} - \frac{1}{a} = d\) \(\frac{1}{c} - \frac{1}{b} = d\)

Express the terms in terms of a common variable

From the two equations above, we can express each term in terms of \(d\): \(\frac{1}{a} = \frac{1}{b} - d\) \(\frac{1}{b} = \frac{1}{c} - d\) Now, let's substitute \(\frac{1}{a}\) into the equation of the line: \[\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = \frac{x}{b-d} + \frac{y}{c-d} + \frac{1}{c} = 0\]

Test each given point to see if it satisfies the equation

Now, let's substitute each point into the equation and see if it satisfies the condition: (a) \((-1, -2)\) -> \(\frac{-1}{b-d} - \frac{2}{c-d} + \frac{1}{c} = 0\) (b) \((-1, 2)\) -> \(\frac{-1}{b-d} + \frac{2}{c-d} + \frac{1}{c} = 0\) (c) \(\left(1, -\frac{1}{2}\right)\) -> \(\frac{1}{b-d} - \frac{1}{2(c-d)} + \frac{1}{c} = 0\) (d) \((1, -2)\) -> \(\frac{1}{b-d} - \frac{2}{c-d} + \frac{1}{c} = 0\) Upon testing each point, we see that the fixed point that satisfies the condition is: \(\boxed{\text{(c) } \left(1, -\frac{1}{2}\right)}\)

Key concepts

Common Difference

Understanding the concept of a common difference is crucial when we're dealing with an arithmetic sequence. This sequence is a list of numbers where the difference between any two successive members is a constant. This constant value is what we refer to as the common difference, denoted as d.

In practical terms, if we have three terms in succession, say a, b, and c, then the common difference would be b - a and also c - b. It’s what makes the sequence 'arithmetic.' If you’re handed a sequence and need to know if it’s arithmetic, just find the differences between consecutive terms. If these differences remain the same throughout, you've indeed got an arithmetic sequence on your hands!

In the problem provided, recognizing that \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\) form an arithmetic sequence allows us to set up two expressions that define this consistent spacing, using the common difference d. Grasping this concept is imperative for solving problems related to arithmetic sequences, and hence the correct identification of the common difference directly influences the solving of such equations.

Equation of a Line

An equation of a line in algebra is a formula that describes all the points along that line. A line is determined by its slope, also known as its steepness, and where it crosses the y-axis, called the y-intercept. The most common form of the equation of a line is the slope-intercept form, which looks like \(y = mx + b\), where m is the slope and b is the y-intercept.

However, lines can be expressed in different forms, such as the standard form \(Ax + By + C = 0\), which is closely related to our problem. When dealing with equations of lines, it's beneficial to understand how to manipulate and work with these forms to insert coordinates, identify slopes, and ascertain where the line crosses the axes.

Our problem involves a more complex line equation: \(\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0\). It's derived from the concept of an arithmetic sequence and pushes us to apply our understanding of lines in a less conventional context. We must rearrange and substitute terms into this line equation to find the fixed point that the line passes through, which is a creative application of basic line equations.

Sequence and Series

The terms 'sequence' and 'series' often appear together in mathematics, but they are not interchangeable. A sequence is essentially a set of numbers in a specific order. These numbers are called the terms of the sequence. A sequence can be finite with a determined end, or infinite where the terms continue indefinitely. Examples of sequences include arithmetic sequences, as mentioned earlier, or geometric sequences where each term is multiplied by a constant to get the next term.

On the other hand, series refers to the sum of the terms of a sequence. If you were to add up the terms of a sequence instead of listing them, you’d get a series. It's the act of combining the terms in a sequence to find their total that transforms a sequence into a series.

Understanding the difference is vital for solving problems, as certain problems might ask us to find a specific term within a sequence, while others may require us to calculate the sum of a series. Both concepts are widely used in various branches of mathematics, including algebra, where they are essential in analyzing patterns, making predictions, and solving problems like the one we’re examining, which uses sequence terms to decode the equation of a line.