Question

Find a pair of equations describing a line passing through the point (-2,-5,1) that is parallel to the \(x\) -axis.

Short answer

Question: Find a pair of equations describing a line passing through the point (-2, -5, 1) that is parallel to the x-axis. Answer: The pair of equations describing the line is: \(x = -2\) \(\frac{y + 5}{k} = \frac{z - 1}{m}\) where k and m are any constants.

Step-by-step solution

Find the Direction Ratios of the Line

Since the line is parallel to the x-axis, it won't have any change in the x-coordinate, which means that the direction ratios are in the form of (0, k, m), where k and m are any constants.

Write the Parametric Equation of the Line

To write the parametric equation of the line, we can use the given point and direction ratios. Since the line passes through point (-2, -5, 1), we have: \(x = -2 + 0t \implies x = -2\) \(y = -5 + kt \implies y = -5 + kt\) \(z = 1 + mt \implies z = 1 + mt\) Here, t is the parameter.

Write the Pair of Cartesian Equations

To write the pair of Cartesian equations, we can eliminate the constant parameter t from the above parametric equations. From the equation for x, we have x = -2, which is one of our equations. We can rewrite the other two equations as: \(t = \frac{y + 5}{k}\) \(t = \frac{z - 1}{m}\) Now, we can equate the two expressions for t to get the second equation: \(\frac{y + 5}{k} = \frac{z - 1}{m}\) So, the required pair of equations describing the line is: \(x = -2\) \(\frac{y + 5}{k} = \frac{z - 1}{m}\) Note: Since the direction ratios (0, k, m) involve arbitrary constants k and m, the second equation will involve these constants as well.

Key concepts

Parametric Equations

Parametric equations are a detailed way to express the coordinates of points that make up a geometric object, such as a line, circle, or surface.
They use a parameter, often denoted as \( t \), to define each point on the object. This parameter varies over a specific interval, allowing us to describe the motion along a path or the position of points.For a line in space, the parametric equations define the x, y, and z coordinates in terms of \( t \):

• \(x = x_0 + a t\)
• \(y = y_0 + b t\)
• \(z = z_0 + c t\)

Here, \((x_0, y_0, z_0)\) is a point on the line, and \((a, b, c)\) are direction ratios indicating the direction of the line.
In our example, the line passes through the point \((-2, -5, 1)\) and is parallel to the x-axis, simplifying the parametrization to \(x = -2\), \(y = -5 + kt\), and \(z = 1 + mt\), as there's no change in the x-coordinate.

Direction Ratios

Direction ratios are a set of three numbers that indicate the direction of a line relative to the coordinate axes. These ratios are proportional to the components of a vector parallel to the line.
For instance, if a line has direction ratios \((a, b, c)\), it can be parallel to a vector that has these same components.

Understanding Parallel Lines

When a line is described as parallel to one of the coordinate axes, like the x-axis, it implies that the change in the other coordinates can be arbitrary while the x-coordinate does not change:

• For parallelism to the x-axis: direction ratios are \((0, k, m)\).
• In our problem, since no change occurs along the x-axis, its direction ratio is zero.

Knowing the direction ratios helps to set the parametric equations, as they demonstrate how a movement in one parameter (\(t\)) shifts along the line.

Cartesian Equations

Cartesian equations involve expressing a line or plane using equations in terms of x, y, and z without any parameters.
Unlike parametric equations that rely on a parameter, Cartesian equations eliminate the parameter to provide a direct correlation between the coordinates.To find the Cartesian equations from parametric equations:

• First, identify the independent variable, which often remains constant, such as \( x = -2 \) in our example.
• Next, equate the other parametric lines to remove the parameter \(t\).

For instance, from:\( y = -5 + kt \) and \( z = 1 + mt \), it follows:\[ t = \frac{y + 5}{k} = \frac{z - 1}{m} \]This equation gives the second Cartesian equation, reflecting the constant relationship of \( y \) and \( z \) with respect to \( t \).
Such forms of Cartesian equations are particularly useful for solving systems of equations and visualizing geometric positions and relationships.