If ` z_1 ` lies on `|z-3| + |z + 3| = 8` such that arg `z_1 = pi//6 `, then ` 37|z_1 |^(2)` = _________.

Answer

448

Explanation

`|z-3|+|z+3| = 8`

Given locus is ellipe `(x^(2))/(a^(2)) +(y^(2))/(b^(2)) =1` with length of major axis
`2a = 8 and e=(6)/(8) =(3)/(4)`
`therefore b^(2) =a^(2) (1-e^(2)) = 7`
Therefore, ellipse is `(x^(2))/(16)+(y^(2))/(7) = 1`
Let `|z_(1)| = r` . The point `P(z_(1))`.
`rArr (rcos.(pi)/(6), rsin.(pi)/(6))`
`therefore (3r^(2))/(64) + (r^(2))/( 28) = 1rArr r^(2) = (448)/(37)`