Locus

1Summary 重點摘要

Learn how to find path of all points satisfying a condition.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The concept of locusA locus is the figure formed by all points satisfying a given condition, and the key word is 'all', meaning the whole set of points rather than one or two. For example, all points at distance 3 from a fixed point form a circle. Why stress 'all'? A correct locus must satisfy two directions: every point meeting the condition lies on the figure, and every point on the figure meets the condition; neither can be dropped. A common mistake is drawing a few qualifying points and calling that the answer, ignoring infinitely many others.
2. Step 2: Locus equidistant from a fixed pointAll points at a fixed distance r from a fixed point P form the circle with centre P and radius r. For example, all points at distance 4 from O form the circle of centre O and radius 4; every point on the circle is exactly 4 from O, points inside are closer and points outside are farther. Why is this the most basic locus? It is exactly the definition of a circle. A common mistake is confusing the circle with its interior; the locus is only the circumference, not the region inside.
3. Step 3: Locus equidistant from two pointsAll points equidistant from two fixed points A and B form the perpendicular bisector of segment AB. This line is perpendicular to AB and passes through its midpoint, and every point on it is equally far from A and B. For example, if A and B are 6 centimetres apart, any point on the bisector has equal distances to A and B, such as the point 3 centimetres from the midpoint. Why does it hold? It can be proved with congruent triangles. A common mistake is drawing only a segment; the locus is an infinite straight line.
4. Step 4: Locus equidistant from two linesAll points equidistant from two intersecting lines form two angle bisectors, which are perpendicular to each other and bisect the two pairs of vertically opposite angles formed by the lines. For example, if two lines meet at 60° and 120°, one bisector splits 60° into 30° and 30°, the other splits 120° into 60° and 60°. Why two lines? Points equidistant from both lines occupy the two pairs of opposite angle regions, one bisector in each region. A common mistake is drawing only one bisector; if the figure spans all four regions, add the second.
5. Step 5: Writing a locus equationTo find a locus equation, let an arbitrary point of the locus be (x, y), then translate the geometric condition into an equation in x and y. For example, the locus of points at distance 5 from (2, 3): the distance formula gives √((x − 2)² + (y − 3)²) = 5, and squaring both sides yields (x − 2)² + (y − 3)² = 25, exactly the standard form of a circle. Why let the point be arbitrary? The equation must hold for every point of the locus. A common mistake is dropping the right side when squaring, or expanding (x − 2)² as x² − 4.
6. Step 6: Equation for equidistance from two pointsExample: find the locus of points equidistant from A(0, 0) and B(4, 0). Let the point be (x, y) and write the equality with squared distances to avoid roots: x² + y² = (x − 4)² + y². Why may squared distances be used? Both distances are non-negative, and equal distances hold if and only if the squares are equal, which avoids the square root and simplifies the algebra. The equation states that the squared distance to A equals that to B. A common mistake is a sign error in expanding; (x − 4)² expands to x² − 8x + 16, and the middle term −8x must not be lost.
7. Step 7: Simplifying and checkingContinue simplifying: from x² = x² − 8x + 16, cancel x² on both sides to get 0 = −8x + 16, then x = 2, so the locus is the line x = 2, precisely the perpendicular bisector of AB since the midpoint of AB is (2, 0). Check: take the point (2, 1) on the line; its distance to A is √(4 + 1) = √5 and its distance to B is also √(4 + 1) = √5, equal as required. Why check? A slip during simplification is caught immediately. A common mistake is cancelling x² as −x², which corrupts the whole equation.

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Locus = all points satisfying a condition
• Equidistant from fixed point → circle
• Equidistant from two points → perpendicular bisector
• Equidistant from two lines → angle bisector
• Let (x, y), write equation based on condition, simplify