Question

The following co-ordinates are:
A=(a, b+c)
B=(b, c+a)
C=(c, a+b)
Then show that these co-ordinates are colinear.....

Answer 1

The points are  A(a,b+c), B(b,c+a), C(c,a+b).

Hey mate,, here is ur ans..

If the area of triangle is zero then the points are called collinear points.

If three points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear then

[ x₁ (y₂ - y₃) + x₂( y₃ - y₁)+  x₃ (y₁ -  y₂) ] = 0.

⇒ [ a( c + a - a - b) + b( a + b - b - c) + c( b + c - c - a) ] = 0

⇒ [ ac - ab + ab - bc + bc - ac ] = 0

= 0.

∴ the points (a,b+c), (b,c+a), (c,a+b) are collinear.

#hope it helps

Answer 2

For pts. to be collinear,

$\left[\begin{array}{ccc}a&b+c&1\\b&c+a&1\\c&a+b&1\end{array}\right]$

⇒a(c+a-a-b) - (b+c)(b-c) + 1(ab +b² - c² - ac)

⇒ac-ab-b²+c²+ab+b²-c²-ac

⇒0

Therefore ,the three pts. ate collinear.

Hope this helps you!!