Match the following lists:

Answer

`a to q; b to p; c to s; d to r;`

Explanation

a. Minimum value if equal to distance between (0,0) and `(cos alpha, sin alpha)` which is equal to.
b. `|z-1|+|z+1| = 2` repesent the line segment joining points '1' and '-1'. `|z+2|+|z-2| = 8` represents the ellipse with foci -2 and 2. Clearly , line segment does not intersect the ellipse.
c.`(z_(1) -3z_(2))(barz_(1) -3barz_(2)) = (3-z_(1)barz_(2))(3-barz_(1)z_(2))`
`rArr |z_(1)|^(2) -|z_(1)|^(2) + 9|z_(2)|^(2) - 9=0`
`rArr (1-|z_(2)|^(2))(|z_(1)|^(2)-9) =0`
`rArr |z_(1)|=3`
d. `z=(2sqrt(3)+2i)/((1-i)^(6)) +((1+i)^(6))/((2sqrt(3) -2i)^(8))`
`= (z_(1)^(8))/(barz_(2)^(6))+(z_(2)^(6))/(barz_(1)^(8)) =(|z_(1)|^(16) +|z_(2)|^(12))/((barz_(2))^(6) (barz_(1))^(8))`
`rArr arg (z) = - 8 arg(barz_(1)) - 6 arg (barz_(2))`
`= 8 arg(z_(1)) + 6 arg(z_(2))`
`=8.(pi)/(6)+ 6.(pi)/(4) = (4pi)/(3) + (3pi)/(2) = (17pi)/(6)`
` therefore |4sin theta| = 4|sin.(17pi)/(6)| = 2`