Question

The Equation of line passing through \((1,2,1)\) and \([(2 \mathrm{x}-1) / 3]=[(1-\mathrm{y}) / 3]=[(3 \mathrm{z}-2) / 5] \mathrm{is}\) (A) \([(2 \mathrm{x}-2) / 3]=[(2-\mathrm{y}) / 3]=[(3 \mathrm{z}-3) / 5]\) (B) \([(2 x+2) / 3]=[(2+y) / 3]=[(3 z+3) / 5]\) (C) \([(2 x-1) /(-3)]=[(1-y) /(-3)]=[(3 z-2) /(-1)]\) (D) None of these

Short answer

The short answer is: (C) \(\frac{2x-1}{-3}=\frac{1-y}{-3}=\frac{3z-2}{-1}\).

Step-by-step solution

Determine direction ratios of the given line

The given line has a parametric equation of the form: \[\frac{x-1}{3} = \frac{y-1}{3} = \frac{z-2}{5}\] Let the above fractions on the right-hand side be equal to \(k\). Then, we have: \[x-1 = 3k\] \[y-1 = 3k\] \[z-2 = 5k\] The direction of this line can be represented by the direction ratios \((3k, 3k, 5k)\). For simplification, we can remove the common factor \(k\) to get the direction ratios \((3, 3, 5)\).

Find the general equation of the line using the given point and direction ratios

Next, we need to find the general equation of a line in three-dimensional space passing through the given point \((1, 2, 1)\) and having direction ratios \((3, 3, 5)\). This can be formulated as: \[\frac{x-1}{3} = \frac{y-2}{3} = \frac{z-1}{5}\]

Compare the general equation to the given options

Now that we have the general equation of the line, we can compare it to the given options to determine which one is correct. (A) \(\frac{2x-2}{3}=\frac{2-y}{3}=\frac{3z-3}{5}\) (B) \(\frac{2x+2}{3}=\frac{2+y}{3}=\frac{3z+3}{5}\) (C) \(\frac{2x-1}{-3}=\frac{1-y}{-3}=\frac{3z-2}{-1}\) (D) None of these Comparing the general equation of the line, \(\frac{x-1}{3} = \frac{y-2}{3} = \frac{z-1}{5}\), to the given options, we can see that it is the same as option (C). Thus, the correct answer is (C) \(\frac{2x-1}{-3}=\frac{1-y}{-3}=\frac{3z-2}{-1}\).

Key concepts

Direction Ratios

In the three-dimensional space, direction ratios provide a way to describe the direction of a line or vector. They are essentially a set of three numbers that are proportional to the components of the directional vector for the line. When you're working with a line in 3D, it's crucial to pinpoint its orientation, and direction ratios do just that.

For example, consider a line that passes through the point \(1, 2, 1)\) and has a direction vector with the components \(3, 3, 5)\). These are the direction ratios, and they tell us that for every unit we move along the x-axis, we also move 3 units along the y-axis and 5 units along the z-axis. This concept is crucial in understanding the nature of the line such as its slope and direction in the space.

To find these ratios, you can use two points on the line or a point and a directional vector. In our exercise, we extrapolated that for the given line, the direction ratios are simplified to \(3, 3, 5)\). These ratios are often given in the form of multiples, which in the context of a line, indicate its direction in the simplest form, devoid of any specific magnitude or length.

Parametric Equation of a Line

The parametric equation of a line is a convenient way to represent a line in three dimensions because it captures both a point on the line and its direction. By introducing a parameter, typically denoted as \(k\) or \(t\), we generate equations that describe every point on the line as the parameter varies.

For a line in 3D space passing through a known point \(x_0, y_0, z_0)\) with direction ratios \(a, b, c)\), the parametric equations become: \[x=x_0+at\], \[y=y_0+bt\], \[z=z_0+ct\].

The parameter \(t\) represents a scalar multiple of the direction vector \(a, b, c)\) and varies along the real numbers to yield every point on the line. In our original problem, the parametric equations derived from the given information were: \[x-1 = 3k\], \[y-1 = 3k\], \[z-2 = 5k\], where \(k\) is the parameter that can take any real value. This form is powerful for its simplicity and ease of visualization in 3D space.

General Equation of a Line

Understanding the general equation of a line in three-dimensional space can seem complex, but it's a consistent way to define a line's position and direction. This equation relates to the parametric form but is often presented without explicit parameters to succinctly define the line. It's typically written as a system of ratios that equate the differences in the line's coordinates to its direction ratios.

Here's what it looks like in its general form: \[\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}\]. The denominators \(a, b, c)\) are the direction ratios while \(x_0, y_0, z_0)\) represent a point known to lie on the line. This equation is very effective as it can summarize the entirety of the line’s path through space using just its direction and a single point.

In our exercise, by plugging in the point \(1, 2, 1)\) and the direction ratios \(3, 3, 5)\), we derived the general equation: \[\frac{x-1}{3} = \frac{y-2}{3} = \frac{z-1}{5}\]. Finally, to solve the original problem, we compared this general equation to the provided options and deduced the correct one.