Q. Prove that the lines (b-c)x+(c-a)y+(a-b)=0, (c-a)x+(a-b)y+(b-c)=0 and (a-b)x+(b-c)y+(c-a)=0 are concurrent.

Asked by imabhi264 | 28th Dec, 2017, 06:31: PM

Expert Answer:

begin mathsize 16px style open vertical bar table row cell straight b minus straight c end cell cell straight c minus straight a end cell cell straight a minus straight b end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close vertical bar
straight R subscript 1 plus straight R subscript 2
open vertical bar table row cell straight b minus straight a end cell cell straight c minus straight b end cell cell straight a minus straight c end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close vertical bar
open vertical bar table row cell negative open parentheses straight a minus straight b close parentheses end cell cell negative open parentheses straight b minus straight c close parentheses end cell cell negative open parentheses straight c minus straight a close parentheses end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close vertical bar
Taking space left parenthesis negative 1 right parenthesis space common space from space straight R subscript 1
left parenthesis negative 1 right parenthesis open vertical bar table row cell open parentheses straight a minus straight b close parentheses end cell cell open parentheses straight b minus straight c close parentheses end cell cell open parentheses straight c minus straight a close parentheses end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close vertical bar
straight R subscript 1 space and space straight R subscript 3 space are space same.
Hence comma
open vertical bar table row cell straight b minus straight c end cell cell straight c minus straight a end cell cell straight a minus straight b end cell row cell straight c minus straight a end cell cell straight a minus straight b end cell cell straight b minus straight c end cell row cell straight a minus straight b end cell cell straight b minus straight c end cell cell straight c minus straight a end cell end table close vertical bar equals 0
Lines space are space concurrent. end style

Answered by Sneha shidid | 31st Dec, 2017, 07:35: PM