If the points with the coordinates (a, ma), `{b,(m+1)b},` {c,(m+2)c}` are collinear, then which one of the following is correct ?

Options

A. a, b, c are in arithmetic progression for all m
B. a, b, c are in geometric progression for all m
C. a, b, c are in harmonic progression for all m
D. a, b, c are in arithmetic progression only for m = 1

Answer

C

Explanation

As given :
Points `A (a,ma),B[b,(m+1)b]"and"C[c,(m+2)c]` are collinear.
`implies" "a{(m+1)b-(m+2)c}+b{(m+2)c-ma}`
`+c{ma-(m+1)b}=0`
`implies" "mab+ab-mac-2ac+mbc+2bc-mab+mac-mbc-bc=0`
`implies" "ab-2ac+2bc-bc=0`
`implies" "ab+bc=2ac`
Dividing both the sides by abc, we get
`(1)/(c)+(1)/(a)=(2)/(b)`
`implies" "(1)/(a),(1)/(b),(1)/(c)` are in A.P`
`implies" "a,b,c` are in Harmonic progression for all m.