Question

Equation of a line whose inclination is \(45^{\circ}\) and making an intercept of 3 units on X-axis is (1) \(x+y-3=0\) (2) \(x-y-3=0\) (3) \(x-y+3=0\) (4) \(x+y+3=0\)

Short answer

Answer: (2) \(x - y - 3 = 0\)

Step-by-step solution

Find the slope from the inclination

We are given that the inclination of the line is \(45^{\circ}\). Recall that the slope (m) of a line can be found using the tangent of the inclination, i.e., \(m = \tan(\theta)\) The slope of the line is given by: \(m = \tan(45^{\circ})\) Since \(\tan(45^{\circ}) = 1\), our slope is: \(m = 1\)

Find the equation of the line using the point-slope equation

We are given that the line makes an intercept of 3 units on the X-axis, which corresponds to the point \((3,0)\). Since we now know the slope of the line (\(m=1\)), we can use the point-slope equation to find the equation of the line: The general point-slope equation is given by: \(y - y_1 = m(x - x_1)\) Substituting the given point \((x_1,y_1) = (3,0)\) and the slope \(m=1\), we get: \(y - 0 = 1(x - 3)\) Simplifying the equation, we get the final equation of the line as: \(x - y - 3 = 0\) Thus, the correct answer is (2) \(x - y - 3 = 0\).

Key concepts

Slope of a Line

Understanding the slope of a line is fundamental in algebra and geometry. It is a measure of how steep a line is and the direction it goes. When looking at a graph, if a line rises as you move from left to right, it has a positive slope. Conversely, if it falls, the slope is negative.

The slope is calculated as the ratio of the 'rise' over 'run' between two points on the line. In mathematical terms, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \(m\) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Let's consider the slope of a line that has an inclination of \(45^{\text{\textdegree}}\). Because we know that the slope of such a line is the tangent of its inclination angle, in this case \( m = \tan(45^{\text{\textdegree}}) = 1 \). This means for every unit the line goes right (run), it also goes up (rise) by the same amount, thus forming a 45-degree angle with the x-axis.

Point-Slope Equation

The point-slope equation is another critical tool used to find the equation of a straight line. It forms a relationship involving the slope of a line and coordinates of a particular point it passes through.

To express this, consider a line with slope \(m\) passing through a point \( (x_1, y_1) \). The point-slope equation is as follows: \( y - y_1 = m (x - x_1) \). This formula allows us to construct the equation of a line when we know its slope and one point it crosses.

For instance, if a line has a slope of 1 and an x-intercept at 3 units, it means the line goes through the point \( (3,0) \). Applying this to the point-slope equation gives us \( y - 0 = 1(x - 3) \), which simplifies to \( x - y - 3 = 0 \) as the line's equation. This equation shows how the line's slope and a single point dictate its graph on a coordinate plane.

Inclination of a Line

The inclination of a line refers to the angle it makes with the positive direction of the x-axis. It's a way of describing the direction of a line in terms of angles, and it can be any value from \(0^{\text{\textdegree}}\) to less than \(180^{\text{\textdegree}}\).

Most importantly, the inclination is directly related to the slope. The slope is the tangent of the inclination angle, symbolized mathematically as \( m = \tan(\theta) \), where \( \theta \) is the inclination angle. For example, a line with a \(45^{\text{\textdegree}}\) inclination with the x-axis represents a slope of 1, as the tangent of \(45^{\text{\textdegree}}\) equals one.

Note that an inclination of \(0^{\text{\textdegree}}\) means the line is horizontal, hence the slope is 0. An inclination of \(90^{\text{\textdegree}}\) means the line is vertical, which is a special case since the slope is undefined (as we cannot divide by zero in the rise over run calculation).