Question 1: Find the distance between the following pairs of points:

(i) (ii)

Answer:

(i)

(ii)

Question 2: Find the distance between the points and having coordinates and

Answer:

Question 3: Using distance formula prove that the following points are collinear:

(i)

(ii)

(iii)

Answer:

(i) Given

Here,

Hence, the points are collinear.

(ii) Given

Here,

Hence, the points are collinear.

(iii) Given

Here,

Hence, the points are collinear.

Question 4: Determine the points in (i) (ii) and (iii) which are equidistant from the points

Answer:

(i) We know that the z-coordinate of every point on the xy-plane is zero.

So let be the point on the xy-plane such that

Now,

Now,

Solving equation i) and ii) simultaneously. we get

(ii) We know that the x-coordinate of every point on the yz-plane is zero.

So let be the point on the xy-plane such that

Now,

Now,

Solving equation i) and ii) simultaneously. we get

(iii) We know that the y-coordinate of every point on the zx-plane is zero.

So let be the point on the xy-plane such that

Now,

Now,

Solving equation i) and ii) simultaneously. we get

Question 5: Determine the point on z-axis which is equidistant from the points

Answer:

Let M be the point on the z-axis.

Then the coordinates of will be

Let M be equidistant from the points

Since

Squaring both sides

Question 6: Find the point on y-axis which is equidistant from the points

Answer:

Let the point of y-axis be which is equidistant from the points

Therefore

Squaring both sides we get

Therefore the required point on the y-axis is

Question 7: Find the points on which are at a distance from the point

Answer:

Let the point of z-axis be

Hence the coordinates of the required point are and

Question 8: Prove that the triangle formed by joining the three points whose coordinates are is an equilateral triangle.

Answer:

Let are the coordinates of the

Now,

Therefore is an equilateral triangle.

Question 9: Show that the points are the vertices of an isosceles right-angled triangle.

Answer:

Let be the coordinates of

Clearly,

Therefore the is a right-angles isosceles triangle.

Question 10: Show that the points are the vertices of a square.

Answer:

Let are the vertices of a quadrilateral ABCD.

Therefore The four sides are equal.

Therefore The diagonals are equal.

Hence is a square.

Question 11: Prove that the point taken in order are the vertices of a parallelogram. Also, show that, is not a rectangle.

Answer:

Let are the vertices of a quadrilateral ABCD.

Therefore Since the opposite sides are equal, therefore, ABCD is a parallelogram.

Therefore The diagonals are not equal.

Hence is not a rectangle.

Question 12: Show that the points are the vertices of a rhombus.

Answer:

Let are the vertices of a quadrilateral ABCD.

Therefore Since all sides are equal, therefore, ABCD is a rhombus.

Question 13: Prove that the tetrahedron with vertices at the points is a regular one.

Answer:

The faces of a regular tetrahedron are equilateral triangles.

Hence this face is an equilateral triangle. Similarly, the other faces are equilateral triangles.

Hence the tetrahedron is a regular one.

Question 14: Show that the points lie on a sphere whose center is Find also its radius.

Answer:

Let lie on a sphere whose center is

Since AP, BP, CP and DP are radii, therefore AP = BP = CP = DP.

Now,

Hence lie on a sphere whose center is and radius is

Question 15: Find the coordinates of the point which is equidistant from the four points

Answer:

Let P(x, y, z) be the point which is equidistant from the four points

The

Similarly, we have

Similarly, we also have

Question 16: If are two points. Find the locus of a point which moves in such away that

Answer:

Let the coordinate of point be

Given:

Squaring both sides we get:

This is the locus of the point

Question 17: Find the locus of if where are the points

Answer:

Let the coordinate of point be

Given:

Squaring both sides we get:

This is the locus of the point

Question 18: Show that the points are the vertices of an equilateral triangle.

Answer:

Let be the vertices of Then,

Therefore is an equilateral triangle.

Question 19: Are the points the vertices of a right-angled triangle?

Answer:

Let the vertices of a

Therefore the points are not vertices of a right angles triangle.

Question 20: Verify the following:

(i) are vertices of an isosceles triangle.

(ii) are vertices of a right-angled triangle.

(iii) are vertices of a parallelogram.

(iv) are the vertices of a rhombus.

Answer:

(i) Let be the vertices of a

Therefore we see

Thus the given points are the vertices of an isosceles triangle.

(ii) Let be the vertices of a

Therefore we see

Thus the given points are the vertices of a right-angled triangle.

(iii) Let be the vertices of a quadrilateral

Therefore we see

Since. each pair of opposite sides are equal. the quadrilateral

(iv) Let be the vertices of a quadrilateral

Therefore we see

Since. each sides are equal. the quadrilateral is a rhombus

Question 21: Find the locus of the points which are equidistant from the points

Answer:

Let be any point that is equidistant from the points

Therefore

Hence the locus is

Question 22: Find the locus of the point, the sum of whose distances from the points is equal to

Answer:

Let P(x, y, z) be any point, the sum of whose distances from the points is equal to

Therefore PA + PB = 10

Squaring both sides we get

Squaring both sides we get

Hence the locus is

Question 23: Show that the points and the vertices of a parallelogram but not a rectangle.

Answer:

Let are the vertices of a quadrilateral ABCD.

Therefore The opposite sides are equal. Therefore ABCD is a parallelogram.

Therefore The diagonals are not equal.

Hence is not a rectangle.

Question 24: Find the equation of the set of the points P such that its distances from the points are equal.

Answer:

Let be any point that is equidistant from the points

Therefore

Squaring both sides we get

Hence, the required equation is