Describe and construct each of the following loci:

(i) The locus of the tip of a minute hand of a watch.

(ii) The locus of the tip of the pendulum of a clock.

(iii) The locus of a point 5 cm from a fixed point O.

(iv) The locus of a point at a distance of 3 cm from a fixed line AB.

(v) The locus of a point equidistant from the arms OA and OB of ∠AOB.

(vi) The locus of the centres of all circles, each of radius 1 cm and touching externally a fixed circle with centre O and radius 3 cm.

(vii) The locus of the centres of all circles to which both the arms of an angle ∠AOB are tangents.

(viii) The locus of a point 1 cm from the circumference of a fixed circle towards the centre O, whose radius is 3 cm.

(ix) The locus of a point 1 cm from the centre of a circle of radius 2.5 cm.

(x) The locus of a stone dropped from a tower.

(xi) AB is a fixed line. State the locus of a point P such that ∠APB = 90°.

(i) Let radius of minute hand be r units.

The minute hand rotates around a fixed center. The distance from the tip to the center remains constant, forming a circle.

Hence, the locus of the tip of a minute hand of a watch is the circumference of a circle with radius equal to the length of minute hand.

(ii) A pendulum swings in an arc from a fixed pivot, always at a constant distance from the pivot.

Hence, the locus of the tip of the pendulum of a clock is an arc of a circle whose radius is equal to length of pendulum.

(iii) All points 5 cm from one point form a circle.

Hence, the locus of a point 5 cm from a fixed point O is the circumference of a circle with center O and radius 5 cm.

(iv) All points 3 cm from a line lie on two lines parallel to the original line.

Hence, the locus of a point at a distance of 3 cm from a fixed line AB are two parallel lines, 3 cm away from AB.

(v) Points equidistant from two intersecting lines lie on the angle bisector.

Hence, the locus of a point equidistant from the arms OA and OB of ∠AOB is the bisector of ∠AOB.

(vi) The locus of centres of all circles of radius 1 cm touching externally a fixed circle of radius 3 cm. Distance between centers = 3 + 1 = 4 cm, forming a concentric circle.

Hence, the locus of the centres of all such circles is a circle with center O and radius 4 cm.

(vii) The locus of centres of all circles tangent to both arms of ∠AOB. The center must be equidistant from both arms i.e. angle bisector of the angle between OA and OB.

Hence, the locus of the centres of all circles to which both the arms of an angle ∠AOB are tangents is the bisector of ∠AOB.

(viii) The locus of a point 1 cm from the circumference of a circle of radius 3 cm, towards the centre will be a circle with same center and radius equal to 2 cm (3 - 1).

Hence, the locus of a point is a circle concentric with the fixed circle, having center O and radius 2 cm.

(ix) The locus of a point 1 cm from the centre of a circle of radius 2.5 cm. The radius of the original circle is irrelevant. Distance from center is fixed at 1 cm.

Hence, the locus of a point 1 cm from the centre of a circle is the circumference of a circle with center O and radius 1 cm.

(x) The locus of a stone dropped from a tower is a straight line parallel to tower.

Hence, the locus of a stone dropped from a tower is a straight vertical line segment.

(xi) We know that the angle in a semi-circle is always equal to 90°.

Hence, the locus of point P will be the circle whose diameter is AB.