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**Distance Between Two Points**

Let P and Q be two given points in space. Let the co-ordinates of the points P and Q be and with respect to a set OX, OY, OZ of rectangular axes.

The position vectors of the points P and Q are given by

Now we have

**B. Section Formula**

(for external division take â€“ve sign)

To determine the co-ordinates of a point R which divides the joining of two points P and internally in the ratio m1 : m2. Let OX, OY, OZ be a set of rectangular axes.

The position vectors of the two given points are given by

Also if the co-ordinates of the point R are (x, y, z), then

Now the point R divides the join of P and Q in the ratio m_{1} : m_{2}, so that

or [Using (1), (2) and (3)]

**Remark :** The middle point of the segment PQ is obtained by putting m_{1} = m_{2}. Hence the co-ordinates of the middle point of PQ are

**Centroid Of A Triangle:**

Let ABC be a triangle. Let the co-ordinates of the vertices A, B and C be and respectively. Let AD be a median of the Î”ABC. Thus D is the mid point of BC.

Now if G is the centroid of Î”ABC, then G divides AD in the ratio 2 : 1. Let the co-ordinates of G be (x, y, z).

**Centroid Of A Tetrahedron :**

Let ABCD be a tetrahedron, the co-ordinates of whose vertices are = 1, 2, 3, 4.

Let be the centroid of the face ABC of the tetrahedron. Then the co-ordinates of are

The fourth vertex D of the tetrahedron does not lie in the plane of Î”ABC. We know from statics that the centroid of the tetrahedron divides the line DG_{1} in the ratio 3 : 1. Let G be the centroid of the tetrahedron and if (x, y, z) are its co-ordinates, then

**Ex.1 P is a variable point and the co-ordinates of two points A and B are (â€“2, 2, 3) and (13, â€“3, 13) respectively. Find the locus of P if 3PA = 2PB.**

**Sol.** Let the co-ordinates of P be (x, y, z).

Now it is given that 3PA = 2PB i.e., 9PA^{2} = 4PB^{2}. ....(3)

Putting the values of PA and PB from (1) and (2) in (3), we get

9{(x + 2)^{2} + (y â€“ 2)^{2} + (z â€“ 3)^{2}} = 4 {(x â€“ 13)^{2} + (y + 3)^{2} + (z â€“ 13)^{2}}

or 9 {x^{2} + y^{2} + z^{2} + 4x â€“ 4y â€“ 6z + 17} = 4{x^{2} + y^{2} + z^{2} â€“ 26x + 6y â€“ 26z + 347}

or 5x^{2} + 5y^{2} + 5z^{2} + 140x â€“ 60 y + 50 z â€“ 1235 = 0

or x^{2} + y^{2} + z^{2} + 28x â€“ 12y + 10z â€“ 247 = 0

This is the required locus of P.

**Ex.2 Find the ratio in which the xy-plane divides the join of (â€“3, 4, â€“8) and (5, â€“6, 4). Also find the point of intersection of the line with the plane.**

**Sol. **Let the xy-plane (i.e., z = 0 plane) divide the line joining the points (â€“3,4, â€“8) and (5, â€“6, 4) in the ratio , in the point R. Therefore, the co-ordinates of the point R are

....(1)

But on xy-plane, the z co-ordinate of R is zero

(4Î¼ - 8) / (Î¼ + 1) = 0, or Î¼ = 2. Hence Î¼ : 1 = 2 : 1. Thus the required ratio is 2 : 1.

Again putting the co-ordinates of the point R become (7/3, â€“8/3, 0).

**Ex.3 ABCD is a square of side length â€˜aâ€™. Its side AB slides between x and y-axes in first quadrant. Find the locus of the foot of perpendicular dropped from the point E on the diagonal AC, where E is the midpoint of the side AD.**

**Sol. **Let vertex A slides on y-axis and vertex B slides on x-axis coordinates of the point A are (0, a sin Î¸) and that of C are (a cos Î¸ + a sin Î¸, a cos Î¸)

...(2)

Form (1) and (2),

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