If $a$ is equals to $9+4\sqrt{5}$ and $b$ is equals to $\frac{1}{a}$, then find the value of $a^2+b^2$.
Given: $a=9+4\sqrt{5}$ and $b=\frac{1}{a}$.
To do: To find the value of $a^2+b^2$.
Solution:
As given, $a=9+4\sqrt{5}$ and $b=\frac{1}{a}=\frac{1}{9+4\sqrt{5}}$
$\Rightarrow b=\frac{1}{9+4\sqrt{5}}=\frac{1}{9+4\sqrt{5}}\times\frac{9-4\sqrt{5}}{9-4\sqrt{5}}$
$\Rightarrow b=\frac{9-4\sqrt{5}}{9^2-( 4\sqrt{5})^2)}$
$\Rightarrow b=\frac{9-4\sqrt{5}}{81-80}$
$\Rightarrow b=9-4\sqrt{5}$
Therefore, $a^2+b^2=( 9+4\sqrt{5})^2+( 9-4\sqrt{5})^2$
$=81+72\sqrt{5}+80+81-72\sqrt{5}+80$
$=322$
Thus, $a^2+b^2=322$.
Related Articles
- Simplify \( (\frac{1}{2})^{4} \div(\frac{-1}{2})^{3} \) is equals to(a.) $\frac{-1}{2}$(b.) $\frac{1}{2}$(c.) 1(d.) 0
- If a = 5 and b = $-$2 then the find the value of:(a $-$ b)2 $-$ (a $+$ b)2
- $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}+\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}=a+b \sqrt{10}$, find a and b.
- If \( \sin \mathrm{A}=\frac{1}{2} \), then the value of \( \cot \mathrm{A} \) is(A) \( \sqrt{3} \)(B) \( \frac{1}{\sqrt{3}} \)(C) \( \frac{\sqrt{3}}{2} \)(D) 1
- Given that $sin\ \theta = \frac{a}{b}$, then $cos\ \theta$ is equal to(A) \( \frac{b}{\sqrt{b^{2}-a^{2}}} \)(B) \( \frac{b}{a} \)(C) \( \frac{\sqrt{b^{2}-a^{2}}}{b} \)(D) \( \frac{a}{\sqrt{b^{2}-a^{2}}} \)
- If a = $\frac{7}{2}$ and b = $-\frac{5}{4}$Find $\frac{a\ +\ b}{a\ -\ b}$.
- If $\frac{1}{b} \div \frac{b}{a} = \frac{a^2}{b}$, where a, b not equal to 0, then find the value of $\frac{\frac{a}{(\frac{1}{b})} - 1}{\frac{a}{b}}$.
- If \( x=\frac{\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}-\sqrt{a^{2}-b^{2}}} \), then prove that \( b^{2} x^{2}-2 a^{2} x+b^{2}=0 \).
- If \( x-\frac{1}{x}=5 \), find the value of(a) \( x^{2}+\frac{1}{x^{2}} \)(b) \( x^{4}+\frac{1}{x^{4}} \)
- If \( a+b=5 \) and \( a b=2 \), find the value of(a) \( (a+b)^{2} \)(b) \( a^{2}+b^{2} \)
- Simplify each of the following products:\( (\frac{1}{2} a-3 b)(3 b+\frac{1}{2} a)(\frac{1}{4} a^{2}+9 b^{2}) \)
- If the lines given by \( 3 x+2 k y=2 \) and \( 2 x+5 y+1=0 \) are parallel, then the value of \( k \) is(A) \( \frac{-5}{4} \)(B) \( \frac{2}{5} \)(C) \( \frac{15}{4} \),b>(D) \( \frac{3}{2} \)
- If \( \tan \theta=1 \) and \( \sin \phi =\frac{1}{\sqrt{2}}, \) then the value of \( \cos (\theta+\phi) \) is:(a) -1(b) 0(c) 1(d) \( \frac{\sqrt{3}}{2} \)
- If \( a^{2}-b^{2}=0 \), then the value of \( \frac{a}{b} \) is____.
- Find the value of $a \times b$ if \( \frac{3+2 \sqrt{3}}{3-2 \sqrt{3}}=a+b \sqrt{3} \).
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
To Continue Learning Please Login