Let `P` denotes a complex number `z=r(costheta+isintheta)` on the Argand's plane, and `Q` denotes a complex number `sqrt(2|z|^(2))(cos(theta+(pi)/(4))+isin(theta+(pi)/(4)))`. If `'O'` is the origin, then `DeltaOPQ` is

Options

A. isosceles but not right angled
B. right angled but not isosceles
C. right isosceles
D. equilateral

Answer

C

Explanation

`(c )`
`Z_(P)=r(costheta+isintheta)`
`Z_(0)=sqrt(2|z|^(2))(cos(theta+(pi)/(4))+isin(theta+(pi)/(4)))`
`=sqrt(2)r[cos(theta+(pi)/(4))+isin(theta+(pi)/(4))]`
From the figure,
`cos"(pi)/(4)=(2r^(2)+r^(2)-x^(2))/(2*sqrt(2r)*r)=(3r^(2)-x^(2))/(2sqrt(2)r^(2))`
`:.1=(3r^(2)-x^(2))/(2r^(2))`
`impliesr^(2)=x^(2)impliesx=rimplies` Triangle is right isosceles.