Locus and Transformations

TL;DR

Locus is the path traced by a point satisfying a given geometric condition, resulting in an equation. Transformations move geometric figures without changing their shape or size. Understanding these concepts helps you describe and manipulate shapes mathematically.

1. The Mental Model

Think of locus as drawing a line or curve by following a rule, like a treasure hunt clue leading you along a specific path. Transformations are like taking that drawing and sliding it, flipping it, or turning it on the page.

2. The Core Material

When a point moves in a plane (or space) such that it always satisfies one or more given conditions, the path it traces is called its locus. Your goal is usually to find the equation that describes this path.

2.1 Finding the Locus Equation

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To find the equation of a locus:
1. Assume the moving point is P(x, y).
2. Write down the given geometric condition(s) using x and y.
3. Simplify the equation you get. This simplified equation is the locus.

Common Distance Formulas you'll use:
* Distance between P(x, y) and A(x1, y1): sqrt((x - x1)^2 + (y - y1)^2)
* Distance from P(x, y) to a line Ax + By + C = 0: |Ax + By + C| / sqrt(A^2 + B^2)

2.2 Introduction to Transformations

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Transformations are operations that move or change a geometric figure. In JEE, you'll mainly deal with rigid transformations, meaning the shape and size of the figure don't change.

Types of Transformations:

• Translation: Sliding a figure in a specific direction without rotation or reflection.
  • If you translate (x, y) by (h, k), the new point is (x+h, y+k).
• Reflection: Flipping a figure over a line (the axis of reflection).
  • Reflection across the x-axis: (x, y) becomes (x, -y)
  • Reflection across the y-axis: (x, y) becomes (-x, y)
  • Reflection across the line y=x: (x, y) becomes (y, x)
  • Reflection across the origin: (x, y) becomes (-x, -y)
• Rotation: Turning a figure around a fixed point (the center of rotation) by a certain angle.
  • Rotation of (x, y) by 90 degrees counter-clockwise about the origin: (-y, x)
  • Rotation of (x, y) by 180 degrees about the origin: (-x, -y)
  • Rotation of (x, y) by 270 degrees counter-clockwise (or 90 degrees clockwise) about the origin: (y, -x)

graph TD
    A["Geometric Condition(s)"] --> B["Assume P(x, y)"]
    B --> C["Apply Condition(s) using (x, y)"]
    C --> D["Formulate Equation"]
    D --> E["Simplify Equation"]
    E --> F["Result: Locus"]

3. Worked Example

Problem: Find the locus of a point P(x, y) such that its distance from the point A(3, 0) is always equal to its distance from the y-axis.

Solution:
1. Let the moving point be P(x, y).
2. Distance from P(x, y) to A(3, 0):
PA = sqrt((x - 3)^2 + (y - 0)^2)
PA = sqrt((x - 3)^2 + y^2)
3. Distance from P(x, y) to the y-axis (which is the line x = 0):
The distance from (x, y) to x = 0 is |x|.
4. According to the condition, PA = |x|:
sqrt((x - 3)^2 + y^2) = |x|
5. Square both sides to remove the square root and absolute value:
(x - 3)^2 + y^2 = x^2
6. Expand and simplify:
x^2 - 6x + 9 + y^2 = x^2
-6x + 9 + y^2 = 0
y^2 = 6x - 9

This is the equation of the locus, which is a parabola.

4. Key Takeaways

• Locus problems involve translating geometric conditions into algebraic equations using P(x, y).
• Always define your moving point as (x, y) at the start.
• Be proficient with basic distance formulas: point-point and point-line.
• Transformations like translation, reflection, and rotation move figures without changing their shape.
• Memorize the standard formulas for reflections across common lines (x-axis, y-axis, y=x) and rotations about the origin.

Common Mistakes to Avoid:
- Forgetting to square both sides when dealing with distances involving sqrt.
- Not handling absolute values correctly when calculating distances to lines.
- Confusing reflection formulas (e.g., reflecting across x-axis vs. y-axis).
- Incorrectly applying rotation formulas, especially for a specific angle.

5. Now Try It

Find the locus of a point P(x, y) such that the sum of its distances from A(2, 0) and B(-2, 0) is always 6. Simplify your result to its standard form. You'll know you're successful if your final equation represents an ellipse.