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What is the Locus of a Point Equidistant from Two Points?
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 points is the path traced by all points that are the same distance away from two fixed points. This path always forms a straight line that perfectly cuts the line segment joining the two fixed points into two equal halves, and is also perpendicular to it.
Simple Example
Quick Example
Imagine you have two friends, Priya and Rahul, standing at different spots on a playground. If you want to stand exactly in the middle so you are the same distance from both Priya and Rahul, you would stand on a specific line. This line is the locus of all points equidistant from Priya and Rahul.
Worked Example
Step-by-Step
Let's find the locus of a point equidistant from Point A (2, 3) and Point B (6, 3).
Step 1: Understand the goal. We need to find the line where every point on it is the same distance from A and B.
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Step 2: Find the midpoint of the line segment AB. The midpoint formula is ((x1+x2)/2, (y1+y2)/2).
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Step 3: Substitute the coordinates: ((2+6)/2, (3+3)/2).
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Step 4: Calculate the midpoint: (8/2, 6/2) = (4, 3). This midpoint is on our locus line.
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Step 5: Find the slope of the line segment AB. The slope formula is (y2-y1)/(x2-x1).
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Step 6: Substitute the coordinates: (3-3)/(6-2) = 0/4 = 0. This means AB is a horizontal line.
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Step 7: The locus line must be perpendicular to AB. If AB is horizontal (slope 0), then the perpendicular line must be vertical. A vertical line passing through (4, 3) has the equation x = 4.
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Step 8: So, the locus of a point equidistant from A(2,3) and B(6,3) is the line x = 4. This is the perpendicular bisector of the segment AB.
Answer: The line x = 4.
Why It Matters
Understanding loci helps in designing efficient networks, like placing mobile towers equidistant from villages for best signal. It's crucial in computer graphics for object placement and in robotics for path planning. Engineers use this concept to design balanced structures and in navigation systems.
Common Mistakes
MISTAKE: Students often confuse the locus with just the midpoint. | CORRECTION: The midpoint is *one* point on the locus, but the locus is the *entire line* of all such points.
MISTAKE: Forgetting that the locus must be perpendicular to the line segment joining the two points. | CORRECTION: Remember, 'equidistant' implies a perpendicular bisector, not just any line passing through the midpoint.
MISTAKE: Only finding the equation of the line segment between the two points. | CORRECTION: The question asks for the locus of points equidistant from the two points, which is a different line (the perpendicular bisector), not the line connecting them.
Practice Questions
Try It Yourself
QUESTION: What is the locus of a point equidistant from Point P(1, 1) and Point Q(1, 5)? | ANSWER: The line y = 3
QUESTION: A new metro station needs to be built so it is exactly the same distance from two busy markets, Market A at (0, 0) and Market B at (4, 0). What is the equation of the line where the station could be located to satisfy this condition? | ANSWER: The line x = 2
QUESTION: Find the equation of the locus of a point equidistant from Point C(2, 1) and Point D(4, 3). | ANSWER: The line y = -x + 5
MCQ
Quick Quiz
What kind of line is the locus of a point equidistant from two fixed points?
A line parallel to the segment joining the points.
The line segment joining the two points.
The perpendicular bisector of the segment joining the points.
A circle passing through the two points.
The Correct Answer Is:
C
The locus of points equidistant from two fixed points is always the perpendicular bisector of the line segment connecting those two points. It's not parallel, nor the segment itself, nor a circle.
Real World Connection
In the Real World
Imagine a new mobile tower needs to be placed in a rural area between two villages, Kanti and Rampur, to provide equal signal strength to both. The engineers will use this concept to find the ideal line where the tower can be placed, ensuring it's equidistant from both villages for fair coverage. This is a common practice in network planning for companies like Jio or Airtel.
Key Vocabulary
Key Terms
LOCUS: The path traced by a point moving according to a specific rule | EQUIDISTANT: Being the same distance from two or more things | PERPENDICULAR BISECTOR: A line that cuts another line segment into two equal halves at a 90-degree angle | MIDPOINT: The exact middle point of a line segment
What's Next
What to Learn Next
Great job understanding loci! Next, you can explore 'Locus of a point equidistant from a line' and 'Locus of a point equidistant from two intersecting lines'. These build on what you've learned and will help you solve even more complex geometry problems.
