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The length of longer diagonal of the p...

The length of longer diagonal of the parallelogram constructed on `5a + 2b` and `a-3b` . If it is given that `|a| = 2sqrt(2),|b|` = 3 and angle between a and b is `(pi)/(4)` is

A

15

B

`sqrt(113)`

C

`sqrt(593)`

D

`sqrt(369)`

Text Solution

Verified by Experts

The correct Answer is:
C
Length of the two diagonasl will be
`d_(1)=|(5a+2b)+(a-3b)|`
and `d_(2)=|(5a+2b)-(a-3b)|`
`impliesd_(1)=|6a-b|,d_(2)=|4a+5b|`
thus, `d_(1)=sqrt(|6a|^(2)+|-b|^(2)+2|6a||-b|cos(pi-pi//4))`
`=sqrt(36(2sqrt(2))^(2)+9+12*2sqrt(2)*3*(-(1)/(sqrt(2))))=15`
`d_(2)=sqrt(|4a|^(2)+|5b|^(2)+2|4a||5b|"cos"(pi)/(4))`
`=sqrt(16xx8+25xx9+40xx2sqrt(2)xx3xx(1)/(sqrt(2)))`
`=sqrt(593)`
`therefore` Length of the longer diagonal`=sqrt(593)`.
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