Question

If three point ( h, 0), ( a, b ) and (0 , k ) lie on a line, show that .

Answer 1

The points ( h,0 ), ( a,b ) and ( 0,k ) are on a line.

The formula for slope m of a non-vertical line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,

m= y 2 − y 1 x 2 − x 1 (1)

Let, m 1 be the slope of line passing through the points ( h,0 ), and ( a,b ).

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( h,0 ) and ( a,b ) respectively in equation (1),

m 1 = b−0 a−h = b a−h

Let, m 2 be the slope of line passing through the points ( a,b ), and ( 0,k ).

Substitute the values of ( x 1 , y 1 ) and ( x 2 , y 2 ) as ( a,b ), and ( 0,k ) respectively in equation (1).

m 2 = k−b 0−a = k−b −a

The given points lie on a straight line, so the slopes of these points are equal.

m 1 = m 2 (2)

Substitute the values of m 1 and m 2 in equation (2).

b a−h = k−b −a −a⋅b=( k−b )⋅( a−h ) −ab=ka−kh−ab+bh ka+bh=kh

Further simplify above equation.

ka kh + bh kh = kh kh a h + b k =1

Hence, a h + b k =1 is proved.