Question

Line \([(x-1) / c]=[(y+3) /(-1)]=[(z-3) / 2]\) and \([(x-3) / 6]=[(y-1) / 3]=[(4-z) / 6]\) if direction are same then \(\mathrm{c}=\) (A) \(-2\) (B) 2 (C) \((1 / 3)\) (D) \(-(1 / 3)\)

Short answer

\(c = -2\) (A)

Step-by-step solution

Compare the direction ratios

The lines will have the same direction if their direction ratios are proportional. Let's compare the direction ratios of the lines. For the first line, the direction ratios are: \((c, -1, 2)\). For the second line, the direction ratios are: \((6, 3, -6)\).

Set up proportions

Now we need to set up proportions to determine if the direction ratios are proportional. We can set up the proportions as: \[\frac{c}{6} = \frac{-1}{3} = \frac{2}{-6}\]

Solve for c

Using the proportions above, we can solve for c: From the first proportion, we have: \[\frac{c}{6} = \frac{-1}{3}\] 3c = -6 So, c = -2. Now let's check if it also satisfies the last proportion: \[\frac{2}{-6} = \frac{-1}{3}\] Indeed, it holds true. Thus, the correct value of c is -2. The answer is (A) \(-2\).

Key concepts

3D Coordinate Geometry

3D coordinate geometry introduces a spatial understanding where every point is defined by three coordinates, typically represented as \(x, y, z\) in a Cartesian coordinate system. Imagine floating in space where you can move left or right, up or down, and forwards or backwards. Each movement corresponds to a change along the x, y, or z-axis, respectively.

In the realm of 3D geometry, lines are often described by equations that highlight their direction in space. These equations use direction ratios to express the relative movement along each of the three axes. When you see an equation like \(\frac{x-1}{c}=\frac{y+3}{-1}=\frac{z-3}{2}\), it's telling us how we travel from a specific point along the line by scaling movement along the axes.

• The numerator in each fraction represents how far we move from a fixed point along the line.
• The denominator indicates the line's direction ratios, mapping out the line's trajectory through three-dimensional space.

Understanding how lines are positioned and how they move through this 3D environment is essential for solving problems involving distances, angles, and intersections between geometric figures.

Proportional Direction Ratios

The concept of proportional direction ratios is crucial when dealing with parallel lines or, similarly, lines that share the same direction in space. Two lines are said to maintain the same direction if their direction ratios are in proportion.

For example, if one line has direction ratios \(a, b, c\) and another has \(k\cdot a, k\cdot b, k\cdot c\), where \(k\) is a scalar, then the lines have the same direction because you can scale one set of direction ratios by \(k\) to obtain the other.

• The key is to establish a proportion, such as \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), where \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\) are the direction ratios of the two lines in question.
• If this proportion holds true for each ratio pair, then the lines either point in the same direction or point in exactly opposite directions.

In practice, confirming that lines share the same direction allows us to solve various geometric problems, such as finding the angle between two lines or determining the point of intersection.

Solving Linear Equations

Solving linear equations is foundational to algebra and understanding geometric concepts. When working with proportions, as is the case with direction ratios, we often set up linear equations to solve for unknown variables.

Returning to our exercise, we used proportions to determine the value of \(c\) by setting up equal ratios and solving the resultant equations:

• We have \(\frac{c}{6} = \frac{-1}{3}\), which leads to the linear equation \(3c = -6\).
• Dividing both sides by 3 gives the solution \(c = -2\).
• To ensure consistency, we also check the ratios \(\frac{2}{-6} = \frac{-1}{3}\), confirming our solution is valid for all proportions.

Knowing how to solve such equations allows us to tackle a wide array of problems, not just in geometry but throughout various disciplines that rely on mathematical reasoning. An equation offers a path to finding unknown values that align with the specified conditions, thus solving the geometric puzzle at hand.