What is the locus of a point that moves on a plane in such a way that its distance from a fixed point is always a constant?

We will discuss the definition of ellipse and how to find the equation of the ellipse whose focus, directrix and eccentricity are given.

An ellipse is the locus of a point P moves on this plane in such a way that its distance from the fixed point S always bears a constant ratio to its perpendicular distance from the fixed line L and if this ratio is less than unity.

An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity.

The constant ratio usually denoted by e (0 < e < 1) and is known as the eccentricity of the ellipse.

If S is the focus, ZZ' is the directrix and P is any point on the ellipse, then by definition

\(\frac{SP}{PM}\) = e

⇒ SP = e ∙ PM

The fixed point S is called a Focus and the fixed straight line L the corresponding Directrix and the constant ratio is called the Eccentricity of the ellipse.

Solved example to find the equation of the ellipse whose focus, directrix and eccentricity are given:

Determine the equation of the ellipse whose focus is at (-1, 0), directrix is 4x + 3y + 1 = 0 and eccentricity is equal to  \(\frac{1}{√5}\).

Solution:

Let S (-1, 0) be the focus and ZZ' be the directrix. Let P (x, y) be any point on the ellipse and PM be perpendicular from P on the directrix. Then by definition

SP = e.PM where e = \(\frac{1}{√5}\).

⇒ SP\(^{2}\) = e\(^{2}\) PM\(^{2}\)

⇒ (x + 1)\(^{2}\) + (y - 0)\(^{2}\) = \((\frac{1}{\sqrt{5}})^{2}[\frac{4x + 3y + 1}{\sqrt{4^{2} + 3^{2}}}]\)

⇒ (x + 1)\(^{2}\) + y\(^{2}\) = \(\frac{1}{25}\)\(\frac{4x + 3y + 1}{5}\)

⇒ x\(^{2}\) + 2x + 1 + y\(^{2}\) = \(\frac{4x + 3y + 1}{125}\)

⇒ 125x\(^{2}\) + 125y\(^{2}\) + 250x + 125 = 0, which is the required equation of the ellipse.

● The Ellipse

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Central University of Venezuela

Scott K.

Algebra

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Related Pages
Loci In Geometry
Loci
More Geometry Lessons

The following diagrams give the locus of a point that satisfy some conditions. Scroll down the page for more examples and solutions.

When a point moves in a plane according to some given conditions the path along which it moves is called a locus. (Plural of locus is loci.).

CONDITION 1:

A point P moves such that it is always m units from the point Q.

Locus formed: A circle with center Q and radius m.

Example:
Construct the locus of a point P at a constant distance of 2 cm from a fixed point Q.

Solution:
Construct a circle with center Q and radius 2 cm.

CONDITION 2:

A point P moves such that it is equidistant form two fixed points X and Y.

Locus formed: A perpendicular bisector of the line XY.

Example:
Construct the locus of point P moving equidistant from fixed points X and Y and XY = 6 cm.

Solution:
Construct a perpendicular bisector of the line XY.

CONDITION 3:

A point P moves so that it is always m units from a straight line AB.

Locus formed: A pair of parallel lines m units from AB.

Example:
Construct the locus of a point P that moves a constant distant of 2 cm from a straight line AB.

Solution:
Construct a pair of parallel lines 2 cm from AB.

CONDITION 4:

A point P moves so that it is always equidistant from two intersecting lines AB and CD.

Locus formed: Angle bisectors of angles between lines AB and CD.

Example:
The following figure shows two straight lines AB and CD intersecting at point O. Construct the locus of point P such that it is always equidistant from AB and CD.

Example:
Construct angles bisectors of angles between lines AB and CD.

Five Fundamental Locus Theorems And How To Use Them

Locus Theorem 1: The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius. Locus Theorem 2: The locus of the points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l. Locus Theorem 3: The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.

Locus Theorem 4: The locus of points equidistant from two parallel lines, l1 and l2, is a line parallel to both l1 and l2 and midway between them.

Example 1:
A treasure map shows a treasure hidden in a park near a tree and a statue. The map indicates that the tree and the stature are 10 feet apart. The treasure is buried 7 feet from the base of the tree and also 5 feet from the base of the stature. How many places are possible locations for the treasure to be buried? Draw a diagram of the treasure map, and indicate with an X each possible location of the treasure.

Example 2:
The distance between the parallel line l and m is 12 units. Point A is on line l. How many points are equidistant from lines l and m and 8 units from point A.

Example 3:
Maria’s backyard has two trees that are 40 feet apart. She wants to place lampposts so that the the posts are 30 feet from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lampposts are possible?

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Five Rules Of Locus Theorem Using Real World Examples

Locus is a set of points that satisfy a given condition. There are five fundamental locus rules. Rule 1: Given a point, the locus of points is a circle. Rule 2: Given two points, the locus of points is a straight line midway between the two points. Rule 3: Given a straight line, the locus of points is two parallel lines. Rule 4: Given two parallel lines, the locus of points is a line midway between the two parallel lines.

Rule 5: Given two intersecting lines, the locus of points is a pair of lines that cut the intersecting lines in half.

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Intersection Of Two Loci

Sometimes you may be required to determine the locus of a point that satisfies two or more conditions. We could do this by constructing the locus for each of the conditions and then determine where the two loci intersect.

Example:
Given the line AB and the point Q, find one or more points that are 3 cm from AB and 5 cm from Q.

Solution:
Construct a pair of parallel lines 3 cm from line AB. Draw a circle with center Q and radius 5 cm.

The points of intersections are indicated by points X and Y.

It means that the locus consists of the two points X and Y.

Example:
Given a square PQRS with sides 3 cm. Construct the locus of a point which is 2 cm from P and equidistant from PQ and PS. Mark the points as A and B.

Solution:
Construct a circle with center P and radius 2 cm. Since PQRS is a square the diagonal PR would be the angle bisector of the angle formed by the lines PQ and PS. The diagonal when extended intersects the circle at points A and B.

Note: A common mistake is to identify only one point when there could be another point which could be found by extending the construction lines or arcs; as in the above examples.

GCSE Maths Exam Questions - Loci, Locus And Intersecting Loci

Examples:

1. Draw (i) the locus of a point that moves so that it is always exactly 4 cm from the fixed point X and

   (ii) the locus of points less than 4 cm from the fixed point X.

2. Draw the locus of points no further than 3 cm from A and no further than 4 cm from B.

3. Draw the locus of a point exactly 3 cm away from straight line AB.

4. Draw (i) the locus of a point equidistant from the points X and Y. (ii) the locus of points closer to the point X than the point Y.

   (iii) the locus of points closer to X than Y but no less than 5 cm from X.

5. Draw the locus of points closer to the line AB than the line BC in the rectangle ABCD.

6. A dog is on a lead tethered to a post in the corner of a garden. The lead is 5 m long. A cat is free to roam all parts of the garden but is not allowed within 3 m of the house by its owner. Show the safe area that the cat can safely roam on the diagram below.

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