1. If  x(b-c)+y(c-a)+z(a-b)=0, then prove that (bz-cy)/(b-c)=(cx-az)/(c-a)=(ay-bx)/(a-b)
  2. If the ratio of arithmetic mean of two numbers x and y to their geometric mean is p:q,then prove that x:y=p+square root of p hat 2 minus q hat 2 end root  :p-square root of p hat 2 minus q hat 2 end root

Asked by tarunkumardas32 | 24th Aug, 2019, 11:42: AM

Expert Answer:

2.
fraction numerator begin display style fraction numerator x plus y over denominator 2 end fraction end style over denominator square root of x y end root end fraction equals p over q
rightwards double arrow fraction numerator straight x plus straight y over denominator 2 square root of xy end fraction equals p over q
A p p l y i n g space c o m p o n e n d e space d i v i d e n d o
rightwards double arrow fraction numerator begin display style straight x plus straight y plus 2 square root of xy end style over denominator begin display style x plus y minus 2 square root of xy end style end fraction equals fraction numerator begin display style p plus q end style over denominator begin display style p minus q end style end fraction
rightwards double arrow fraction numerator begin display style open parentheses square root of x begin display style plus end style begin display style square root of y end style close parentheses squared end style over denominator open parentheses square root of x begin display style minus end style square root of y close parentheses squared end fraction equals fraction numerator begin display style p plus q end style over denominator begin display style p minus q end style end fraction
rightwards double arrow fraction numerator begin display style square root of x plus square root of y end style over denominator square root of x minus square root of y end fraction equals fraction numerator begin display style square root of p plus q end root end style over denominator begin display style square root of p minus q end root end style end fraction
A g a i n space a p p l y i n g space c o m p o n e n d e space d i v i d e n d o
rightwards double arrow fraction numerator begin display style square root of x plus square root of y plus square root of x minus square root of y end style over denominator square root of x begin display style plus end style square root of y minus square root of x plus square root of y end fraction equals fraction numerator begin display style square root of p plus q end root plus square root of p minus q end root end style over denominator begin display style square root of p plus q end root minus square root of p minus q end root end style end fraction
rightwards double arrow fraction numerator square root of x over denominator square root of y end fraction equals fraction numerator begin display style square root of p plus q end root plus square root of p minus q end root end style over denominator begin display style square root of p plus q end root minus square root of p minus q end root end style end fraction
rightwards double arrow x over y equals fraction numerator p plus q plus p minus q plus 2 square root of p squared minus q squared end root over denominator p plus q plus p minus q minus 2 square root of p squared minus q squared end root end fraction
rightwards double arrow fraction numerator begin display style x end style over denominator begin display style y end style end fraction equals fraction numerator begin display style 2 p plus 2 square root of p squared minus q squared end root end style over denominator begin display style 2 p minus 2 square root of p squared minus q squared end root end style end fraction
rightwards double arrow fraction numerator begin display style x end style over denominator begin display style y end style end fraction equals fraction numerator begin display style p plus square root of p squared minus q squared end root end style over denominator begin display style p minus square root of p squared minus q squared end root end style end fraction

Answered by Renu Varma | 5th Sep, 2019, 12:18: PM