What is Locus Equidistant from Parallel Lines? | Simple Explanation for Class 8

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What is the Locus of a Point Equidistant from Two Parallel Lines?

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The locus of a point equidistant from two parallel lines is a third straight line that lies exactly halfway between the two original parallel lines. Imagine a path where every step you take keeps you the same distance from both walls of a corridor; that path is the locus.

Simple Example

Quick Example

Think of two parallel train tracks. If a drone flies exactly in the middle, always keeping the same distance from both tracks, the path it traces in the sky is a straight line. This line is the locus of all points equidistant from the two train tracks.

Worked Example

Step-by-Step

Let's say we have two parallel lines. Line 1 is y = 2 and Line 2 is y = 8.
---Step 1: Understand what 'equidistant' means. It means the same distance from both lines.
---Step 2: Find the distance between the two parallel lines. The distance between y=2 and y=8 is 8 - 2 = 6 units.
---Step 3: A point equidistant from both lines must be half of this distance from each line. Half of 6 units is 3 units.
---Step 4: So, the locus will be 3 units away from y=2 and also 3 units away from y=8.
---Step 5: Starting from y=2, move 3 units upwards: y = 2 + 3 = 5.
---Step 6: Starting from y=8, move 3 units downwards: y = 8 - 3 = 5.
---Step 7: Both calculations give y = 5. This means the locus is the line y = 5.
---Answer: The locus of a point equidistant from the lines y=2 and y=8 is the line y=5.

Why It Matters

Understanding loci helps in designing efficient systems, like planning robotic movements in a factory or setting up Wi-Fi networks for best coverage. Engineers use this concept to ensure fair distribution or optimal placement, impacting fields like Computer Science and even city planning.

Common Mistakes

MISTAKE: Students sometimes think the locus is a single point or a curved line. | CORRECTION: For two parallel lines, the locus of equidistant points is always a straight line, parallel to the original lines.

MISTAKE: Calculating the distance incorrectly by just adding the y-values if one is negative. | CORRECTION: Always find the absolute difference between the y-coordinates (or x-coordinates for vertical lines) to get the positive distance.

MISTAKE: Placing the locus line too close to one of the original lines. | CORRECTION: The locus line must be exactly in the middle, meaning its distance to Line 1 is equal to its distance to Line 2.

Practice Questions

Try It Yourself

QUESTION: What is the equation of the locus of a point equidistant from the lines x = 3 and x = 7? | ANSWER: x = 5

QUESTION: Find the locus of a point equidistant from the lines y = -4 and y = 10. | ANSWER: y = 3

QUESTION: Two parallel roads are represented by the lines 3x + 4y = 12 and 3x + 4y = 22. What is the equation of the path exactly in the middle of these roads? (Hint: The general form for the mid-parallel line between Ax+By=C1 and Ax+By=C2 is Ax+By=(C1+C2)/2) | ANSWER: 3x + 4y = 17

MCQ

Quick Quiz

If two parallel lines are given by y = 1 and y = 9, what is the equation of the locus of a point equidistant from them?

y = 4

y = 5

y = 8

x = 5

The Correct Answer Is:

The distance between y=1 and y=9 is 8 units. Half of this distance is 4 units. So, the locus line will be 4 units from y=1 (which is y=1+4=5) and 4 units from y=9 (which is y=9-4=5).

Real World Connection

In the Real World

Imagine a farmer wants to install a new irrigation pipe exactly in the middle of two parallel rows of crops to ensure equal water distribution. The path for this pipe would be the locus of points equidistant from the two crop rows. Similarly, in urban planning, placing a bus stop exactly between two parallel roads ensures it's equally accessible.

Key Vocabulary

Key Terms

LOCUS: A set of all points that satisfy a given condition | EQUIDISTANT: At an equal distance | PARALLEL LINES: Lines that are always the same distance apart and never meet | EQUATION OF A LINE: A mathematical rule that describes all points on a straight line

What's Next

What to Learn Next

Great job understanding this! Next, explore 'Locus of a point equidistant from two intersecting lines'. This will help you see how the concept of locus changes when the lines are no longer parallel, building a stronger foundation for geometry.