# NCERT Solutions class-11 Maths Exercise 12.2

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Exercise 12.2

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

(i) (2, 3, 5) and (4, 3, 1)

(ii) and

(iii) and

(iv) and

Ans. (i) Let A (2, 3, 5) and B (4, 3, 1) be two points, then

AB =

= = units

(ii) Let A (–3, 7, 2) and B (2, 4, –1) be two points, then

AB =

= = units

(iii) Let A (–1, 3, –4) and B (1, –3, 4) be two points, then

AB =

= = units

(iv) Let A (2, –1, 3) and B (–2, 1, 3) be two points, then

AB =

= = units

2. Show that the points and are collinear.

Ans. Let A (–2, 3, 5), B (1, 2, 3) and C (7, 0, –1) be three points, then

AB =

= = units

BC =

= = units

AC = =

= = units

Here, AC = AB + BC

Therefore A, B and C are collinear.

3. Verify the following:

(i) and are the vertices of an isosceles triangle.

(ii) and are the vertices of right angled triangle.

(iii) and are the vertices of a parallelogram.

Ans. (i) Let A (0, 7, –10) B (1, 6, –6) and C (4, 9, –6) be three vertices of , then

AB =

= = units

BC =

= = units

AC = =

= = units

Here, AB = BC

Therefore is an isosceles triangle.

(ii) Let A (–1, 2, 1), B (–1, 6, 6) and C (–4, 9, 6) be three vertices of , then

AB =

= = units

BC =

= = units

AC = =

= = units

Here, AC2 = AB2 + BC2

Therefore is a right angled triangle.

(iii) Let A (0, 7, 10), B (1, –2, 5), C (4, –7, 8) and D (2, –3, 4) be four vertices of a quadrilateral ABCD, then

AB =

= = units

BC =

= = units

CD =

= = units

= = units

AC =

= = units

BD =

= = units

Here, AB = CD, BC = AD and AC BD

Therefore A, B, C and are the vertices of a parallelogram ABCD.

4. Find the equation of the set of points which are equidistant from the point (1, 2, 3) and

Ans. Let A be any point which is equidistant from points B (1, 2, 3) and C Then

According to question, AB = AC

Squaring both sides, we get

5. Find the equation of the set of points P, the sum of whose distance from A (4, 0, 0) and B is equal to 10.

Ans. Let P be any point, then

According to question, PA + PB = 10

Squaring both sides, we get

Again squaring both sides, we get

This is the required equation.

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