Advertisements
Advertisements
Question
If \[x = 7 + 4\sqrt{3}\] and xy =1, then \[\frac{1}{x^2} + \frac{1}{y^2} =\]
Options
64
134
194
1/49
Advertisements
Solution
Given that `x=7+4sqrt3`, `xy = 1`
Hence y is given as
`y=1/x`
`1/x = 1/(7+4sqrt3)`
We need to find `1/x^2+ 1/y^2`
We know that rationalization factor for `7+4sqrt3` is `7-4sqrt3`. We will multiply numerator and denominator of the given expression `1/(7+4sqrt3)`by,`7-4sqrt3` to get
`1/x = 1/(7+4sqrt3) xx (7-4sqrt3)/(7-4sqrt3)`
`= (7-4sqrt3)/((7)^2 (4sqrt3)^2) `
` = (7-4sqrt3)/(49 - 48)`
` = 7-4sqrt3`
Since `xy=1`so we have
`x=1/y`
Therefore,
`1/x^2 + 1/y^2 = ( 7 - sqrt3)^2 + (7+4sqrt3)^2`
` = 7^2 + (4sqrt3)^2 - 2 xx 7 xx 4sqrt3 + 7 ^2 +(4 sqrt3)^2 + 2 xx 7 xx 4sqrt3`
`= 49 + 48 - 14 sqrt3 + 49 +48 +14sqrt3`
`= 194`
APPEARS IN
RELATED QUESTIONS
Prove that:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`
Show that:
`1/(1+x^(a-b))+1/(1+x^(b-a))=1`
Solve the following equation:
`8^(x+1)=16^(y+2)` and, `(1/2)^(3+x)=(1/4)^(3y)`
Simplify:
`(x^(a+b)/x^c)^(a-b)(x^(b+c)/x^a)^(b-c)(x^(c+a)/x^b)^(c-a)`
Show that:
`((a+1/b)^mxx(a-1/b)^n)/((b+1/a)^mxx(b-1/a)^n)=(a/b)^(m+n)`
If `a=x^(m+n)y^l, b=x^(n+l)y^m` and `c=x^(l+m)y^n,` Prove that `a^(m-n)b^(n-l)c^(l-m)=1`
The value of \[\left\{ 2 - 3 (2 - 3 )^3 \right\}^3\] is
Which of the following is (are) not equal to \[\left\{ \left( \frac{5}{6} \right)^{1/5} \right\}^{- 1/6}\] ?
When simplified \[( x^{- 1} + y^{- 1} )^{- 1}\] is equal to
The simplest rationalising factor of \[\sqrt{3} + \sqrt{5}\] is ______.
