## EXERCISE 12.4

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#### Question 1:

Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex.

#### Answer:

The three vertices of a parallelogram ABCD are given as A (3, –1, 2), B (1, 2, –4), and C (–1, 1, 2). Let the coordinates of the fourth vertex be D (*x*, *y*, *z*).

We know that the diagonals of a parallelogram bisect each other.

Therefore, in parallelogram ABCD, AC and BD bisect each other.

∴Mid-point of AC = Mid-point of BD

⇒ *x* = 1, *y* = –2, and *z* = 8

Thus, the coordinates of the fourth vertex are (1, –2, 8).

#### Question 2:

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

#### Answer:

Let AD, BE, and CF be the medians of the given triangle ABC.

Since AD is the median, D is the mid-point of BC.

∴Coordinates of point D == (3, 2, 0)

Thus, the lengths of the medians of ΔABC are.

#### Question 3:

If the origin is the centroid of the triangle PQR with vertices P (2*a*, 2, 6), Q (–4, 3*b*, –10) and R (8, 14, 2*c*), then find the values of *a*, *b* and *c*.

#### Answer:

It is known that the coordinates of the centroid of the triangle, whose vertices are (*x*_{1}, *y*_{1}, *z*_{1}), (*x*_{2}, *y*_{2}, *z*_{2}) and (*x*_{3}, *y*_{3}, *z*_{3}), are.

Therefore, coordinates of the centroid of ΔPQR

It is given that origin is the centroid of ΔPQR.

Thus, the respective values of *a*, *b*, and *c* are

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#### Question 4:

Find the coordinates of a point on *y*-axis which are at a distance offrom the point P (3, –2, 5).

#### Answer:

If a point is on the *y*-axis, then *x*-coordinate and the *z*-coordinate of the point are zero.

Let A (0, *b*, 0) be the point on the *y*-axis at a distance of from point P (3, –2, 5). Accordingly,

Thus, the coordinates of the required points are (0, 2, 0) and (0, –6, 0).

#### Question 5:

A point R with *x*-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[**Hint** suppose R divides PQ in the ratio *k*: 1. The coordinates of the point R are given by

#### Answer:

The coordinates of points P and Q are given as P (2, –3, 4) and Q (8, 0, 10).

Let R divide line segment PQ in the ratio *k*:1.

Hence, by section formula, the coordinates of point R are given by

It is given that the* x*-coordinate of point R is 4.

Therefore, the coordinates of point R are

#### Question 6:

If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set of points P such that PA^{2} + PB^{2} = *k*^{2}, where *k* is a constant.

#### Answer:

The coordinates of points A and B are given as (3, 4, 5) and (–1, 3, –7) respectively.

Let the coordinates of point P be (*x*, *y*, *z*).

On using distance formula, we obtain

Now, if PA^{2} + PB^{2} = *k*^{2}, then

Thus, the required equation is.