What is the Equation of a Locus?

S6-SA1-0242

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of a locus is a mathematical rule that describes the path traced by a point moving under specific conditions. It's like finding a formula that tells you exactly where a moving object can be, based on certain rules it follows.

Simple Example

Quick Example

Imagine you are holding a piece of chalk and drawing a circle on a blackboard. The chalk tip is a moving point. The 'condition' is that it always stays the same distance from the center of the circle (your elbow, perhaps!). The equation of this locus would be the formula for a circle, showing all possible points the chalk tip can touch.

Worked Example

Step-by-Step

Let's find the equation of the locus of a point P(x, y) that is always 5 units away from the origin O(0, 0).

Step 1: Understand the condition. The distance between point P(x, y) and the origin O(0, 0) is always 5.
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Step 2: Use the distance formula. The distance between two points (x1, y1) and (x2, y2) is sqrt((x2-x1)^2 + (y2-y1)^2).
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Step 3: Apply the distance formula to P(x, y) and O(0, 0). Distance PO = sqrt((x-0)^2 + (y-0)^2).
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Step 4: Set the distance equal to the given condition. sqrt(x^2 + y^2) = 5.
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Step 5: To remove the square root, square both sides of the equation. (sqrt(x^2 + y^2))^2 = 5^2.
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Step 6: Simplify the equation. x^2 + y^2 = 25.

Answer: The equation of the locus is x^2 + y^2 = 25.

Why It Matters

Understanding locus equations helps engineers design safe bridges and buildings, ensuring all parts stay in their intended paths. In robotics, it helps program robots to move precisely. Even game developers use it to create realistic movements for characters and objects in video games.

Common Mistakes

MISTAKE: Forgetting to square both sides after using the distance formula, leaving a square root in the final equation. | CORRECTION: Always square both sides of the equation to eliminate the square root and simplify the expression into a standard algebraic form.

MISTAKE: Not correctly applying the distance formula, especially with negative coordinates, leading to sign errors. | CORRECTION: Double-check the subtraction of coordinates (x2-x1) and (y2-y1) and ensure squaring negative numbers results in positive numbers.

MISTAKE: Confusing the variables (x, y) of the moving point with fixed coordinates given in the problem. | CORRECTION: Always use (x, y) for the general moving point whose locus you are trying to find, and use specific numerical values for any fixed points given.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the locus of a point P(x, y) that is always 3 units away from the point A(1, 2). | ANSWER: (x-1)^2 + (y-2)^2 = 9

QUESTION: A point P(x, y) moves such that its distance from the point (0, 0) is equal to its distance from the x-axis. Find the equation of its locus. (Hint: Distance from x-axis is |y|). | ANSWER: x^2 = 0 (This simplifies to x=0, meaning the y-axis, but the standard form is x^2 = 0 or x^2 = 4ay if it was a parabola). Let's re-evaluate. If distance from (0,0) is sqrt(x^2+y^2) and distance from x-axis is |y|. Then sqrt(x^2+y^2) = |y|. Squaring both sides: x^2+y^2 = y^2. So x^2 = 0. This is the y-axis. Corrected answer: x^2 = 0.

QUESTION: Find the equation of the locus of a point P(x, y) such that its distance from point A(2, 0) is twice its distance from point B(-1, 0). | ANSWER: x^2 + y^2 + 4x + 4 = 0 (This is a circle with center (-2, 0) and radius 0, so it's just the point (-2,0). Let's re-check. PA^2 = 4PB^2. (x-2)^2 + y^2 = 4 * ((x+1)^2 + y^2). x^2 - 4x + 4 + y^2 = 4(x^2 + 2x + 1 + y^2). x^2 - 4x + 4 + y^2 = 4x^2 + 8x + 4 + 4y^2. 3x^2 + 12x + 3y^2 = 0. x^2 + 4x + y^2 = 0. This is a circle. Corrected Answer: x^2 + 4x + y^2 = 0)

MCQ

Quick Quiz

Which of the following represents the locus of a point P(x, y) that is equidistant from the points A(0, 0) and B(4, 0)?

x = 2

y = 2

x^2 + y^2 = 16

x = 4

The Correct Answer Is:

If a point is equidistant from A(0,0) and B(4,0), it lies on the perpendicular bisector of the line segment AB. The midpoint of AB is (2,0), and the line perpendicular to the x-axis passing through (2,0) is x=2.

Real World Connection

In the Real World

Think about how a drone (like those used for food delivery by Swiggy or Zomato) flies. It needs to follow a specific path while maintaining a certain distance from obstacles or its delivery target. The equations of locus help map out these precise flight paths, ensuring safe and efficient deliveries across our cities.

Key Vocabulary

Key Terms

Locus: The path traced by a moving point under certain geometric conditions. | Origin: The point (0, 0) on a coordinate plane. | Distance Formula: A formula used to find the distance between two points in a coordinate system. | Equidistant: Being at the same distance from two or more points or objects.

What's Next

What to Learn Next

Now that you understand locus, you can explore specific types of loci like circles, parabolas, ellipses, and hyperbolas. These concepts are super important for understanding how satellites orbit Earth or how light reflects in a telescope!