7.4 Equations of Loci
A locus is the geometric path or curve traced out by a moving point satisfying one or more given mathematical conditions. In daily life, this concept is seen when an aircraft moves at a constant distance from an air traffic control radar antenna, forming a circular path on the screen, or when an animal tied to a stake walks around with the tether fully taut. By using coordinate geometry and the distance formula, the path of such moving points can be expressed precisely as an algebraic equation.
Learning Objectives
- Define and explain the concept of the locus of a moving point.
- Determine the equation of the locus of a moving point whose distance from a fixed point is constant.
- Determine the equation of the locus of a moving point whose ratio of distances from two fixed points is constant.
- Identify the geometric shape of a locus as a circle or a straight line (perpendicular bisector).
- Apply the concepts of loci to solve geometric and real-world problems.
Types of Loci Formed by Moving Points
Classify whether a locus is a circle or a straight line based on the given geometric condition.
One Fixed Point
\(PA = r\)
The locus is a circle with centre \(A(x_1, y_1)\) and radius \(r\).
Two Fixed Points
Ratio \(m \neq n\): \(PA : PB = m : n\)
The locus is a circle (Circle of Apollonius).
Ratio \(m = n\): \(PA = PB\)
The locus is the perpendicular bisector of line segment \(AB\) — a straight line.
Short Explanation
Use the distance formula to translate a geometric condition into an algebraic equation.
1. Concept of a Locus
A locus is the path taken by a moving point subject to certain defined conditions. Let the moving point be \(P(x, y)\) and apply the distance formula.
2. Constant Distance from One Fixed Point
\(PA = r\)
\(\sqrt{(x - x_1)^2 + (y - y_1)^2} = r\)
Squaring both sides gives a circle: \((x - x_1)^2 + (y - y_1)^2 = r^2\), where \(r > 0\).
3. Constant Ratio of Distances from Two Fixed Points
\(\dfrac{PA}{PB} = \dfrac{m}{n}\)
Squaring both sides gives:
\(n^2\left[(x - x_1)^2 + (y - y_1)^2\right] = m^2\left[(x - x_2)^2 + (y - y_2)^2\right]\)
4. Two Cases Based on the Ratio \(m : n\)
When \(m \neq n\), the locus is a circle (Circle of Apollonius).
When \(m : n = 1 : 1\), the \(x^2\) and \(y^2\) terms cancel out, leaving a linear equation \(ax + by + c = 0\) — the perpendicular bisector of \(AB\).
| Condition | Equation / Relationship | Geometric Shape |
|---|---|---|
| Distance from one fixed point is constant | \((x - x_1)^2 + (y - y_1)^2 = r^2\) | Circle |
| Ratio \(PA : PB = m : n\), \(m \neq n\) | \(n^2\left[(x - x_1)^2 + (y - y_1)^2\right] = m^2\left[(x - x_2)^2 + (y - y_2)^2\right]\) | Circle (Circle of Apollonius) |
| Ratio \(PA : PB = 1 : 1\) | \((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\) | Perpendicular bisector (straight line) |
| Right angle \(\angle ASB = 90^\circ\) | \(m_{SA} \times m_{SB} = -1\) or \(SA^2 + SB^2 = AB^2\) | Circle with diameter \(AB\) |
Steps for Solving Locus Problems
Let the moving point be \(P(x, y)\).
Translate the geometric condition into an algebraic statement using the distance formula.
Square both sides to eliminate all radical signs. Remember to square both numerator and denominator of any ratio.
Expand all squared binomials, group like terms, simplify, and write the final equation equal to zero.
Distance formula
\(\text{Distance} = \sqrt{(x - x_1)^2 + (y - y_1)^2}\)Use this to express the given condition.
Binomial expansion
\((x - a)^2 = x^2 - 2ax + a^2\)Expand carefully before simplifying.
Try to Answer First
Answer in your mind, then press "Check Answer".
Find the equation of the locus of a moving point \(P\) such that its distance from the point \(A(4, -3)\) is always \(6\) units.
\(\sqrt{(x - 4)^2 + [y - (-3)]^2} = 6\)
Square both sides: \((x - 4)^2 + (y + 3)^2 = 36\)
Expand: \(x^2 - 8x + 16 + y^2 + 6y + 9 = 36\)
Simplify: \(x^2 + y^2 - 8x + 6y - 11 = 0\)
Point \(P\) moves such that its distances from \(A(-2, 0)\) and \(B(0, 4)\) are equal. Find the equation of the locus of \(P\).
\(\sqrt{[x - (-2)]^2 + (y - 0)^2} = \sqrt{(x - 0)^2 + (y - 4)^2}\)
Square both sides: \((x + 2)^2 + y^2 = x^2 + (y - 4)^2\)
Expand: \(x^2 + 4x + 4 + y^2 = x^2 + y^2 - 8y + 16\)
Cancel \(x^2\) and \(y^2\): \(4x + 8y - 12 = 0\)
Divide each term by \(4\): \(x + 2y - 3 = 0\)
Point \(P(x, y)\) moves such that its distance from \(S(1, 2)\) and \(T(4, -1)\) is in the ratio \(2 : 1\). Find the equation of the locus of \(P\).
\(\dfrac{\sqrt{(x - 1)^2 + (y - 2)^2}}{\sqrt{(x - 4)^2 + (y + 1)^2}} = \dfrac{2}{1}\)
Square both sides: \(\dfrac{(x - 1)^2 + (y - 2)^2}{(x - 4)^2 + (y + 1)^2} = \dfrac{4}{1}\)
Cross-multiply: \((x - 1)^2 + (y - 2)^2 = 4\left[(x - 4)^2 + (y + 1)^2\right]\)
Expand and simplify: \(x^2 + y^2 - 10x + 4y + 21 = 0\)
Common Mistakes
- Forgetting to square the ratio values. When squaring \(\dfrac{PA}{PB} = \dfrac{2}{1}\), students often write \(\dfrac{2}{1}\) instead of \(\dfrac{4}{1}\), leading to completely incorrect coefficients.
- Sign errors with negative coordinates. For point \(A(4, -3)\), writing \([y - (-3)]^2\) as \((y - 3)^2\) instead of \((y + 3)^2\).
- Incorrect binomial expansion. Writing \((x - 4)^2\) as \(x^2 - 16\) or \(x^2 + 16\) instead of the correct three-term expansion \(x^2 - 8x + 16\).
- Omitting the brackets when cross-multiplying. Writing \(4 \times x^2 - 8x \dots\) without enclosing the entire expanded expression in parentheses, leading to multiplying only the first term by 4.
- Failing to simplify to simplest form. Leaving an equation like \(3x^2 + 3y^2 - 30x + 12y + 63 = 0\) without dividing every term by the common factor 3.
Concept Misunderstandings
Every locus equation must contain \(x^2\) and \(y^2\), so a locus is always a circle.
When distances from two fixed points are equal (\(1 : 1\)), the \(x^2\) and \(y^2\) terms cancel out, forming a straight line — the perpendicular bisector.
If the ratio of distance from \(A\) to \(B\) is \(m : n\), setting up \(\dfrac{PB}{PA} = \dfrac{m}{n}\) is the same as \(\dfrac{PA}{PB} = \dfrac{m}{n}\).
The order of points in the ratio matters. Using \(\dfrac{PB}{PA}\) instead of \(\dfrac{PA}{PB}\) will produce an incorrect locus.
\((x, y)\) represents one specific coordinate on the locus.
\((x, y)\) represents an infinite set of coordinates that satisfy a single governing equation.
Summary
- The locus of a moving point is its path under specified geometric constraints.
- A point moving at a fixed distance \(r\) from a single point \((x_1, y_1)\) produces a circle: \((x - x_1)^2 + (y - y_1)^2 = r^2\).
- A point moving such that its distances from two points have a constant ratio \(m : n\) (\(m \neq n\)) produces a circle.
- A point moving equidistant from two points (\(m : n = 1 : 1\)) produces a straight line, which is the perpendicular bisector of the line segment connecting the two points.
- Always square both sides, expand binomials fully, distribute constants carefully, and simplify coefficients to lowest terms.
Short Activity
Objective: Identify the geometric shape and equation of a locus from its given condition.
A. Objective Quiz
1 What is the geometric shape of the locus of a point \(P(x, y)\) that moves such that its distance from point \(A(2, 3)\) is always equal to its distance from point \(B(6, 9)\)?
2 A point \(P(x, y)\) moves such that its distance from the origin \((0, 0)\) is always \(5\) units. What is the equation of the locus of \(P\)?
B. Fill in the Blanks
3 When the ratio of the distances of a moving point from two fixed points is \(1 : 1\), the resulting locus is the ________ bisector of the line segment joining the two points.
4 If a point \(P(x, y)\) moves such that its distance from point \(A(x_1, y_1)\) is always a constant value \(r\), the equation of its locus is \((x - x_1)^2 + (y - y_1)^2 = \) ________.
C. Matching / Drag and Drop
Drag each condition to the correct equation or geometric shape. On a phone, tap a condition first, then tap the matching box.
