Loci — Multiple Choice Questions

The locus of a point which moves in a plane in such a way that its distance from a fixed point is always constant, is known as :

1. a square

2. an angle

3. a circle

4. a line segment

Answer

The locus of a point that moves in a plane such that its distance from a fixed point is always constant, is known as a circle.

The fixed point is the center of the circle, and the constant distance is the radius of the circle.

Hence, option 3 is the correct option.

The locus of a point which is equidistant from two given fixed points, is the ______ of the line segment joining the given fixed points.

1. median

2. angle bisector

3. altitude

4. perpendicular bisector

Answer

The perpendicular bisector of a line segment is the set of all points that maintain equal distance from the two endpoints of that segment.

Hence, option 4 is the correct option.

The locus of a point which is equidistant from two intersecting lines is the ______ formed by the given lines.

1. pair of lines bisecting the angles

2. pair of parallel lines

3. pair of perpendicular lines

4. none of these

Answer

Let AB and CD meet at point O.

For two intersecting lines, the points that are equidistant from both lines lie on the lines that bisect the angles formed at their intersection, every point maintains equal perpendicular distance from the two intersecting lines.

Hence, option 1 is the correct option.

The locus of a toy bird fixed at the tip of one of the blades of a rotating ceiling fan is a:

1. straight line

2. circle

3. semi-circular arc

4. diameter of the circle so formed

Answer

Given,

The toy bird, fixed at the tip of the blade, is at a constant distance from this central point.

The locus of a point equidistant from a fixed point is a circle with fixed point as centre.

Thus, the locus of toy bird will be a circle.

Hence, option 2 is the correct option.

If A and B are two fixed points, then the locus of a point P such that ∠APB = 90°, is the:

1. square

2. rectangle

3. circle

4. rhombus

Answer

All points P that satisfy ∠APB = 90° lie on a circle for which AB is the diameter.

This is because a right angle is subtended only by the diameter of a circle.

So, the locus is a circle.

Hence, option 3 is the correct option.

A point P moves such that its distance from a fixed line AB is always the same. What is the relation between AB and the path traveled by P ?

1. It is a set of two lines perpendicular to AB.

2. It is a set of two lines parallel to AB drawn on either side at equal distance from it.

3. It is a set of two concentric circles.

4. It is a set of two intersecting lines with AB bisecting the angle between them.

Answer

If a point moves so that its distance from a fixed line remains constant, it must stay on two lines that maintain that same constant distance on both sides of the given line. These lines are parallel to the original line AB.

So the path is two parallel lines, each at the given fixed distance from AB.

Hence, option 2 is the correct option.

A point P moves so that its perpendicular distance from two given parallel lines AB and CD are equal. Then, the locus of the point P is:

1. a line l anywhere in between AB and CD and parallel to them.

2. a line l perpendicular to both AB and CD.

3. a line l in the midway of AB and CD and parallel to them.

4. none of these

Answer

If the perpendicular distances of a moving point from two parallel lines are equal, then the point must lie exactly halfway between them.

The set of all such points forms a single line parallel to both AB and CD, drawn exactly midway between them.

Hence, option 3 is the correct option.

The locus of a point equidistant from two concentric circles is:

1. a circle concentric with the given circles and inside the smaller circle.

2. a circle concentric with the given circles and outside the larger circle.

3. a circle concentric with the given circles and midway between them.

4. the common centre of the two circles.

Answer

For concentric circles, every point’s distance from each circle is measured along the same radial direction because the centre is common.

A point that is equidistant from both circles must lie at a radius that is exactly midway between the two radii.

Hence, option 3 is the correct option.

The locus of a point which is equidistant from a given circle consists of:

1. a pair of circles concentric with the given circle.

2. a circle concentric with the given circle and inside it.

3. a circle concentric with the given circle and outside it.

4. a pair of lines parallel to each other on either side of the centre.

Answer

If a point stays at a constant distance (let k) from a given circle, then:

One set of such points lies outside the circle at distance k, forming a circle of radius (R + k).

Another set lies inside the circle at distance k, forming a circle of radius (R − k).

Hence, option 1 is the correct option.

A and B are two fixed points in a plane. Then, the locus of a point P which moves in such a way that PA² + PB² = AB², is:

1. a square with AB as one of its sides.

2. a rectangle with AB as one of its sides.

3. a rhombus with AB as one of its diagonal.

4. the circumference of a circle with AB as its diameter.

Answer

The condition PA² + PB² = AB² is exactly the Pythagorean relation.

This holds only when ∠APB = 90°.

All points that subtend a right angle at AB lie on a circle with AB as its diameter.

Therefore, the locus of point P is the circle having AB as its diameter.

Hence, option 4 is the correct option.

The locus of the tip of the pendulum of a clock is:

1. a circle

2. a chord of a circle

3. an arc of a circle

4. a diameter of a circle

Answer

Let the fixed pivot be O and the constant length be "r"

A pendulum swings back and forth from a fixed pivot. Its length remains constant, so its tip moves along a portion of a circle, not the full circle.

Therefore, the path traced is an arc of a circle.

Hence, option 3 is the correct option.

The circumcentre of a triangle is the point which is ∶

1. at equal distance from the three sides of the triangle.

2. at equal distance from the three vertices of the triangle.

3. the point of intersection of the three medians.

4. the point of intersection of the three altitudes of the triangle.

Answer

The circumcenter of a triangle is defined as the point equidistant from the three vertices of the triangle.

Hence, Option 2 is the correct option.