If a and b are positive quantities, `(a gt b)` find minimum positive value of `(a sectheta- b tantheta)`

Options

A. `1/(sqrt(a^(2)-b^(2)))`
B. `1/(sqrt(a^(2)+b^(2)))`
C. `sqrt(a^(2)+b^(2))`
D. `sqrt(a^(2)-b^(2))`

Answer

D

Explanation

Let `y=a sec theta -b tan theta`
`implies(dy)/( d theta)a sec theta tan theta -b sec^(2) theta`
Put `(dy)/(d theta) =0impliessec theta (a tan theta-b sec theta) =0`
`impliessin theta=b/a [ :' sec theta 1=0]`
Now `(d^(2)y)/(d theta^(2))gt0`, at `sin theta=b/a`
`:. ` Minimum value is
`y= a . a/(sqrt(a^(2) -b^(2)))- b . b/(sqrt(a^(2)-b^(2)))=sqrt(a^(2)-b^(2))`
`:' sin theta =b/a`
`sec theta =a/(sqrt(a^(2)-b^(2)))`
and `tan theta=b/(sqrt(a^(2)-b^(2)))`