Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
+4 votes
12.7k views
in Mathematics by (63.2k points)
edited by

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A ± sqrt[(A +G )( A−G )] .

1 Answer

+7 votes
by (13.2k points)
selected by
 
Best answer

Solution:

It is given that A and G are A.M. and G.M. between two positive numbers. Let these two positive numbers be a and b.

∴ A M space equals space A space equals space fraction numerator a space plus space b over denominator 2 end fraction space space space space space space space space space space space space... left parenthesis 1 right parenthesisG M space equals space G space equals space square root of a b end root space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

From (1) and (2), we obtain

a + b = 2A … (3)

ab = G2 … (4)

Substituting the value of a and b from (3) and (4) in the identity (a – b)2 = (a + b)2 – 4ab, we obtain

(a – b)2 = 4A2 – 4G2 = 4 (A2G2)

(a – b)2 = 4 (A + G) (A – G)

open parentheses a minus b close parentheses space equals space 2 square root of open parentheses A plus G close parentheses open parentheses A minus G close parentheses end root space space space space space space space space space space... left parenthesis 5 right parenthesis

From (3) and (5), we obtain

2 a space equals space 2 A space plus space 2 square root of open parentheses A plus G close parentheses open parentheses A minus G close parentheses end rootrightwards double arrow a space equals space A space plus space square root of open parentheses A space plus space G close parentheses open parentheses A minus G close parentheses end root

Substituting the value of a in (3), we obtain

b space equals space 2 A space minus A space minus space square root of open parentheses A space plus space G close parentheses open parentheses A minus G close parentheses end root space equals space A space minus space square root of open parentheses A plus G close parentheses open parentheses A minus G close parentheses end root

Thus, the two numbers are A plus-or-minus square root of open parentheses A plus G close parentheses open parentheses A minus G close parentheses end root.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...