- Books Name
- AMARENDRA PATTANAYAK Mathmatics Book

- Publication
- KRISHNA PUBLICATIONS

- Course
- CBSE Class 11

- Subject
- Mathmatics

**Distance between 2 points, section formula**

**Distance between Two Points**

If P (x1, y1, z1) and Q (x2, y2, z2) are the two points, then the distance between P and Q is given by:

Now PA = y2 – y1, AN = x2 – x1 and NQ = z2 – z1

Hence PQ^{2} = (x2 – x1)^{2} + (y2 – y1)^{ 2}+ (z2 – z1)^{ 2}

Therefore PQ = √ [(x2 – x1)^{2} + (y2 – y1)^{ 2}+ (z2 – z1)^{ 2 }]This gives us the distance between two points (x1, y1, z1) and (x2, y2, z2).

In particular, if x1 = y1 = z1 = 0, i.e., point P is origin O, then OQ = √(x2^{2} + y2^{2} + z2^{2})

which gives the distance between the origin O and any point Q (x2, y2, z2).

**Section Formula**

**Example**** : **Find the equation of set of points P such that PA^{2} + PB^{2} = 2k^{2}, where A and B are the points (3, 4, 5) and (–1, 3, –7), respectively.

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

Here PA^{2} = (x – 3)^{ 2} + (y – 4)^{ 2} + ( z – 5)^{ 2}

PB^{2} = (x + 1)^{ 2 }+ (y – 3)^{ 2} + (z + 7)^{ 2}

By the given condition PA^{2} + PB^{2} = 2k^{2}, we have

(x – 3)^{ 2 }+ (y – 4)^{ 2} + (z – 5)^{ 2} + (x + 1)^{ 2} + (y – 3)^{ 2} + (z + 7)^{ 2} = 2k^{2}

i.e., 2x^{2} + 2y^{2} + 2z^{2 }– 4x – 14y + 4z = 2k^{2} – 109.

** **Let the two given points be P(x1, y1, z1) and Q (x2, y2, z2). Let the point R (x, y, z) divide PQ in the given ratio m : n internally

To determine the coordinates of the point P, the following steps are followed:

- Draw AL, PN, and BM perpendicular to XY plane such that AL || PN || BM as shown above.
- The points L, M and N lie on the straight line formed due to the intersection of a plane containing AL, PN and BM and XY- plane.
- From point P, a line segment ST is drawn such that it is parallel to LM.
- ST intersects AL externally at S, and it intersects BM at T internally.

Since ST is parallel to LM and AL || PN || BM, therefore, the quadrilaterals LNPS and NMTP qualify as parallelograms.

Also, ∆ASP ~∆BTP therefore,

Rearranging the above equation we get,

**Sectional Formula (Internally)**

Thus, the coordinates of the point P(x, y, z) dividing the line segment joining the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) in the ratio m:n internally are given by:

**Sectional Formula (Externally)**

If the given point P divides the line segment joining the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) externally in the ratio m:n, then the coordinates of P are given by replacing n with –n as:

This represents the section formula for three dimension geometry.

If the point P divides the line segment joining points A and B internally in the ratio k:1, then the coordinates of point P will be

What if the point P dividing the line segment joining points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) is the midpoint of line segment AB?

In that case, if P is the midpoint, then P divides the line segment AB in the ratio 1:1, i.e. m=n=1. Coordinates of point P will be given as:

Therefore, the coordinates of the midpoint of line segment joining points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are given by,

If the point R divides PQ externally in the ratio m : n, then its coordinates are obtained by replacing n by – n so that coordinates of point R will be

The above procedure can be repeated by drawing perpendiculars to XZ and YZ- planes to get the x and y coordinates of the point P that divides the line segment AB in the ratio m:n internally.

**Sectional Formula (Internally)**

Thus, the coordinates of the point P(x, y, z) dividing the line segment joining the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) in the ratio m:n internally are given by:

**Sectional Formula (Externally)**

If the given point P divides the line segment joining the points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) externally in the ratio m:n, then the coordinates of P are given by replacing n with –n as:

This represents the section formula for three dimension geometry.

If the point P divides the line segment joining points A and B internally in the ratio k:1, then the coordinates of point P will be

What if the point P dividing the line segment joining points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) is the midpoint of line segment AB?

In that case, if P is the midpoint, then P divides the line segment AB in the ratio 1:1, i.e. m=n=1.

Coordinates of point P will be given as:

Therefore, the coordinates of the midpoint of line segment joining points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) are given by,

If the point R divides PQ externally in the ratio m : n, then its coordinates are obtained by replacing n by – n so that coordinates of point R will be