The difference of two numbers is `4`.If the difference of their reciprocals is `4/21`, find the numbers

Options

A. `3, 7 and -3, -7`
B. `5, 9 and -5, -9`
C. `7, 11 and -7, -11`
D. `9, 13 and -9, -13`

Answer

A

Explanation

Let the larger number be x and the smaller number be y .
Then , ` x- y = 4 " "`… (i)
And , ` (1)/( y) - (1)/(x) = ( 4)/( 21)" " [ because x gt y rArr (1)/( y) gt (1)/(x)]`
`rArr (x - y )/( xy ) = ( 4)/( 21)`
` rArr ( 4) /( xy ) = ( 4)/( 21) rArr xy = 21 " "`[using (i)]
` therefore ( x+ y ) = sqrt ((x- y ) ^(2) + 4xy) `
` " " = sqrt( 4^(2) + 4 xx 21 ) = sqrt ( 16 + 84 ) = sqrt (100) = pm 10 `
Thus, we have
`{:(x- y = 4,,... (i)),(x+y= 10,,... (ii)):}} or {{:(x- y = 4,,... (iii)),(x+y = - 10,, ... (iv)):}`
On solving (i) and (ii), we get x = 7 and y = - 3
On solving (iii) and (iv), we get x = - 3 and y = - 7.
Hence, the required numbers are ( 7 and 3) or (-3 and - 7).