ICSE Class 10 Maths Chapter 18 Loci

Chapter 18

Loci

Points To Remember

1. Locus is the path traced out by a moving point which moves according to some given geometrical conditions.

Thus, (i) Every point which satisfies the given geometrical conditions will lie on the locus. And, (ii) Every point lying on the locus will satisfy the given geometrical conditions.

Remark: The plural of locus is loci, read as 'losai'.

Example 1. A circle with centre O and radius r cm is the locus of a point which moves in a plane in such a way that its distance from the fixed point O is always equal to r cm.

Example 2. Let a point P move in such a way that its distance from a fixed line AB is always equal to d cm.

Clearly, the locus of the moving point is a pair of straight lines CD and EF, each parallel to AB at a distance of d cm from it.

Points Equidistant From Two Given Points

Theorem 1. The locus of a point which is equidistant from two given fixed points, is the perpendicular bisector of the line segment joining the given fixed points.

Proof: We shall prove the theorem in two parts (a) and (b) given below.

Part (a): Every point which is equidistant from two fixed points A and B, lies on the perpendicular bisector of AB.

Given: Two fixed points A and B and P is a point, such that PA = PB.

To Prove: P lies on the perpendicular bisector of AB.

Construction: Join AB. Find its middle point M and Join MP.

Hence, P lies on the perpendicular bisector of AB.

Part (b): Conversely, every point on the perpendicular bisector of AB is equidistant from A and B.

Given: Two fixed points A and B, MQ is the perpendicular bisector of AB and P is any point on MQ.

To Prove: PA = PB.

Construction: Join PA and PB.

Proof.

Hence, the locus of a point which is equidistant from two fixed points A and B, is the perpendicular bisector of AB.

Point Equidistant From Two Intersecting Lines

Theorem 2. The locus of a point which is equidistant from two intersecting lines is the pair of lines bisecting the angles formed by the given lines.

Proof: We shall prove the theorem in two parts; Part (a) and Part (b).

Part (a): Every point which is equidistant from two intersecting lines, lies on the bisector of the angle between the given lines.

Given: Two straight lines, AB and CD, intersecting at a point O and P is a point in the interior of ∠BOD such that PM - AB, PN - OD and PM = PN.

To Prove: P lies on the bisector of ∠BOD.

Construction: Join OP.

Proof:

∴ P lies on the bisector of ∠BOD

Thus, P lies on the bisector of ∠BOD

Similarly, if P' is a point such that P'M' - OA and P'N' - OD and P'M' = P'N', then P' lies on the bisector of ∠AOD.

Part (b): Conversely, every point on the angle bisector of two intersecting lines, is equidistant from the lines.

Given: Two lines AB and CD intersecting at a Point O; OE is the bisector of ∠BOD and P is a point on OE ; PM - OB and PN - OD.

To Prove: PM = PN

Proof.

∴ P is equidistant from OB and OD, and therefore from AB and CD.

Similarly, if P' is a point on the bisector of ∠AOD and P'M' - LOA, P'N' - OD, then P'M' = P'N'.

Some Examples Of Loci

In each figure, we shall show the locus by dotted lines.

1. The locus of a point which is equidistant from two given points A and B is the perpendicular bisector of AB.

2. The locus of a point equidistant from two intersecting lines AB and BC is the bisector of ∠ABC.

3. If A and B are two fixed points, then the locus of a point P such that ∠APB = 90° is the circle with AB as diameter. (∴ Angle in a semi-circle is always a right angle)

4. The locus of a point P inside a circle such that PA = PB, where A and B are two fixed points on the circle is a diameter of the circle bisecting the chord AB.

5. The locus of mid-points of all chords parallel to a chord AB of a circle is the diameter of the circle which is the right bisector of AB.

6. The locus of mid-points of all equal chords of a circle concentric with the given circle with radius equal to distance of each chord from the centre.

7. The locus of centres of circles touching a given line AB at a given point T on it is the straight line perpendicular to AB at T.

Teacher's Note

Understanding loci helps us in navigation systems and GPS technology, where we track the path of moving objects in real-time by applying geometrical principles similar to those learned here.

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