Question
Download Solution PDFIf \(x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}\) and \(y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}\) then the value of \(x^2 - xy + y^2 \over x^2 +xy + y^2\) = ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}\) and \(y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}\)
Calculations:
According to question,
\(x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}\)
⇒x = 9 - 4√5
⇒x² = 161 - 72√5
and \(y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}\)
⇒y = 9 + 4√5
⇒y² = 161 + 72√5
Now,
\(x^2 - xy + y^2 \over x^2 +xy + y^2\)
= (161 - 72√5 - 1 + 161 + 72√5)/(161 - 72√5 + 1 + 161 + 72√5)
= (322 - 1)/(322 + 1)
= 321/323
Hence, The Required value is 321/323.
Alternate Method
We know, if x = \(\frac{(p - q)}{(p + q)}\) and y = \(\frac{(p + q)}{(p - q)}\) then (x + y) = \(\frac{2\times (p^2 + q^2)}{(p^2 - q^2)}\)
\(x = {{\sqrt5 - \sqrt4} \over {\sqrt5 + \sqrt4}}\) and \(y = {{\sqrt5 + \sqrt4} \over {\sqrt5 - \sqrt4}}\) so, x + y = \(\frac{2(\sqrt5^2 + \sqrt4^2)}{(5 - 4)}\) = 18 and xy = 1
So, \(x^2 - xy + y^2 \over x^2 +xy + y^2\) = \(\frac{(x + y)^2 - 3xy}{(x + y)^2 -xy}\) = \(\frac{18^2 - 3}{18^2 -1}\) = 321/323
Last updated on Jun 13, 2025
-> The SSC CGL Notification 2025 has been released on 9th June 2025 on the official website at ssc.gov.in.
-> The SSC CGL exam registration process is now open and will continue till 4th July 2025, so candidates must fill out the SSC CGL Application Form 2025 before the deadline.
-> This year, the Staff Selection Commission (SSC) has announced approximately 14,582 vacancies for various Group B and C posts across government departments.
-> The SSC CGL Tier 1 exam is scheduled to take place from 13th to 30th August 2025.
-> Aspirants should visit ssc.gov.in 2025 regularly for updates and ensure timely submission of the CGL exam form.
-> Candidates can refer to the CGL syllabus for a better understanding of the exam structure and pattern.
-> The CGL Eligibility is a bachelor’s degree in any discipline.
-> Candidates selected through the SSC CGL exam will receive an attractive salary. Learn more about the SSC CGL Salary Structure.
-> Attempt SSC CGL Free English Mock Test and SSC CGL Current Affairs Mock Test.
-> Candidates should also use the SSC CGL previous year papers for a good revision.