**Equation of the line passing through the point ** **and** :

**Ref: Mathematics for Joint Entrance Examination JEE (Advanced), Second Edition, Algebra, G Tewani.**

There are two forms of this equation, as given below:

and

**Proof:**

Let and . Let A and B be the points representing and respectively.

Let be any point on the line joining A and B. Let . Then , and . Points P, A, and B are collinear.

See attached JPEG figure 1.

The figure shows that the three points A, P and B are collinear.

Shifting the line AB at the origin as shown in the figure; points O, P, Q are collinear. Hence,

or

is purely real.

or, call this as **Equation 1.**

. Call this as **Equation 2.**

Hence, from (2), if points , , are collinear, then

.

Equation (2) can also be written as

let us call this **Equation 3.**

where and , which in turn equals

, which is a real number.

**Slope of the given line**

In Equation (3), replacing z by , we get ,

Hence, the slope

Equation of a line parallel to the line is (where is a real number).

Equation of a line perpendicular to the line is (where is a real number).

**Equation of a perpendicular bisector**

Consider a line segment joining and . Let the line L be its perpendicular bisector. If be any point on L, then we have (see attached fig 2)

or

or

or

Here, and

**Distance of a given point from a given line:**

(See attached Fig 3).

Let the given line be and the given point be . Then,

Replacing z by in the given equation, we get

Distance of from this line is

which in turn equals

which is equal to finally

.

More later,

Nalin Pithwa