OP Malhotra Class 10 Maths Solutions Chapter 13 Loci Exercise 13 (A)

S Chand Class 10 ICSE Maths Solutions Chapter 13 Loci Ex 13(a)

Question 1. What is the locus of a marble dropped from your hand?
Answer: The locus of the marble will be a vertical line. This path is due to gravity pulling the marble straight down towards the Earth's center.
In simple words: When you drop a marble, it falls straight down. Its path is a straight up-and-down line.

🎯 Exam Tip: Remember that gravity makes objects fall in a straight line, which is why the locus is vertical.

Question 2. What is the locus of the tip of a minute hand of a watch?
Answer: The locus of the tip of a minute hand of a watch is a circle. The minute hand rotates around the center of the watch, keeping a constant distance from it, which forms a perfect circle.
In simple words: The pointy end of the minute hand on a clock moves around in a perfect circle.

🎯 Exam Tip: Any object that rotates around a fixed point at a constant distance will trace a circular path.

Question 3. What is the locus of the tip of the pendulum of a clock ?
Answer: The locus of the tip of the pendulum of a clock is an arc of a circle. A pendulum swings back and forth, always staying the same distance from its pivot point, but it does not complete a full circle.
In simple words: The very end of a clock's pendulum makes a curved line, which is part of a circle.

🎯 Exam Tip: An arc is a portion of a circle, which accurately describes the limited swing of a pendulum.

Question 4. What is the locus of the tip of the nose of a man sitting in a merry go round ?
Answer: The locus of the tip of the nose of a man sitting in a merry-go-round is a circle. As the merry-go-round spins, every part of the man, including his nose, moves in a circular path around the center of rotation.
In simple words: When a man rides a merry-go-round, his nose moves in a round shape, like a circle.

🎯 Exam Tip: Remember that any point on a rotating object, if it maintains a fixed distance from the center of rotation, will trace a circle.

Question 5. What is the locus of the centre of a wheel rolling on a horizontal plane ?
Answer: The locus of the center of a wheel rolling on a horizontal plane is a straight line. The center of the wheel moves horizontally without changing its height, creating a path parallel to the ground.
In simple words: The middle point of a wheel rolling on flat ground moves in a straight line.

🎯 Exam Tip: The center of a rolling wheel always stays at the same height above the ground, leading to a horizontal straight line path.

Question 6. One end of a rope is tied to a donkey and the other end to a peg fixed in a grassy land. What is the locus of the end of the rope tied to the donkey if it grazes the land by keeping the rope fully stretched?
Answer: The locus of the end P of the rope tied to the donkey will be a circle whose radius is the length of the rope. The donkey keeps the rope stretched, meaning its distance from the fixed peg is constant, which defines a circle.
In simple words: If a donkey pulls a rope tight around a fixed stick, the tip of the rope where the donkey is will draw a circle on the ground. The length of the rope is the circle's radius.

🎯 Exam Tip: This is a classic example of how a constant distance from a fixed point defines a circle.

Question 7. What is the locus of mark on a see-saw?
Answer: The locus of a mark P on a see-saw will be an arc of a circle. The see-saw pivots around a central point, so any point on its plank moves along a curved path, forming part of a circle.
In simple words: A mark on a see-saw will move in a curved path, like a piece of a circle, as the see-saw goes up and down.

🎯 Exam Tip: Understand that the fixed pivot point of the see-saw acts as the center for the arc created by any point on the plank.

Question 8. What is the locus of your toe when in gymnasium you climb up a rope smoothly without giving any jerk to your body?
Answer: The locus of your toe P while climbing up a rope smoothly will be a straight line. If you climb without jerking, your body and toe move purely vertically upwards, making a straight path.
In simple words: When you climb a rope without wiggling, your toe moves straight up, drawing a straight line.

🎯 Exam Tip: The key here is "smoothly without any jerk," which means no horizontal movement, only vertical.

Question 9. What is the locus of a point 4 cm from a fixed point O?
Answer: The locus of a point P which remains 4 cm away from a fixed point O is a circle of radius 4 cm. Any point that stays a constant distance from a fixed point will always form a circular path.
In simple words: All the points that are exactly 4 cm away from one special point will form a circle with a radius of 4 cm.

🎯 Exam Tip: This is the fundamental definition of a circle: all points equidistant from a center.

Question 10. What is the locus of a point 1 cm from the circumference of a circle towards the centre, whose radius is 2.5 cm?
Answer: The locus of a point P which is 1 cm from the circumference of a circle with center O and radius 2.5 cm is a circle of radius \( 2.5 - 1 = 1.5 \) cm and with center O. This means it is a concentric circle, sharing the same center as the original.
In simple words: Imagine a big circle with a middle point. If you find all points 1 cm inside the edge of that big circle, they will form a smaller circle with the same middle point. The new circle's radius will be 1.5 cm.

🎯 Exam Tip: For points "towards the center," you subtract the given distance from the original radius. For points "away from the center," you would add it.

Question 11. What is the locus of a point 1 cm from the centre of a circle of radius 2.5 cm ?
Answer: The locus of a point P which is 1 cm from the center O of a given circle of 2.5 cm radius is a circle with center O, meaning it is a concentric circle. This new circle will have a radius of exactly 1 cm.
In simple words: For a circle with center O and a 2.5 cm radius, all points exactly 1 cm away from its center O will form a smaller circle with a 1 cm radius, sharing the same center O.

🎯 Exam Tip: A fixed distance from a point defines a circle; a fixed distance from a circumference requires careful addition or subtraction of radii.

Question 12. What is the locus of all points 1 cm from the bottom of this page ?
Answer: The locus of all points 1 cm from the bottom of this page will be a straight line parallel to the bottom of the page. This line will always be at a constant distance of 1 cm from the bottom edge.
In simple words: If you find all the spots on a page that are exactly 1 cm away from the bottom edge, they will form a straight line that runs across the page, parallel to the bottom.

🎯 Exam Tip: A constant distance from a line always results in a parallel line (or two parallel lines, if not restricted to one side).

Question 13. What is the locus of all points equidistant from the sides of this page ?
Answer: The locus of all the points which are equidistant from the sides of this page will be a straight line parallel to the sides and running mid-way between the sides. This line represents the exact middle between the left and right edges.
In simple words: All the points that are the same distance from both the left side and the right side of a page will form a straight line right down the middle of the page.

🎯 Exam Tip: The locus of points equidistant from two parallel lines is a single line exactly in the middle of them.

Question 14. What is the locus of a point equidistant from two parallel lines?
Answer: The locus of a point P which is equidistant from two parallel lines is a straight line parallel to the given parallel lines and running midway between them. This is the only path where the distance to both original lines remains equal.
In simple words: If you have two straight lines that never cross, all the points that are exactly in the middle of those two lines will form another straight line that is also parallel to them.

🎯 Exam Tip: Always draw a simple diagram to visualize such problems; it clarifies the concept of equidistance.

Question 15. What is the locus of a point, equidistant from the arms of a given angle AOB?
Answer: The locus of a point P which is equidistant from the arms OA and OB of a given angle AOB is the bisector of the angle AOB. This line divides the angle into two equal parts, ensuring any point on it is the same distance from both arms.
In simple words: If you have an angle, the line that cuts the angle exactly in half is where all the points are that are the same distance from both sides of the angle.

🎯 Exam Tip: The angle bisector is a crucial concept in geometry, always representing points equidistant from the two arms of an angle.