Question

Show that points are collinear

Answer 1

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

Three points are collinear means they lie on the same line. So, the area of triangle for these points is zero.

The formula used to determine the area of triangle is,

Δ= 1 2 | x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 |

Here,

x 1 =a y 1 =b+c

x 2 =b y 2 =c+a

x 3 =c y 3 =a+b

Substitute the values in the above formula.

Δ= 1 2 | a b+c 1 b c+a 1 c a+b 1 |

Apply column operation C 1 → C 2 + C 1 .

Δ= 1 2 | a+b+c b+c 1 b+c+a c+a 1 c+a+b a+b 1 | = ( a+b+c ) 2 | 1 b+c 1 1 c+a 1 1 a+b 1 |

From the determinant property, if any two rows or columns are identical, then value of the determinant is zero.

In the above determinant, columns C 1 and C 3 are identical. Therefore, area of the given points is 0 square units.

Hence, points A, B and C are collinear.