Let complex numbers α and 1/(α ¯) lie on circles x - x 0^2 + y - y 0^2 = r^2 , and x - x 0^2 + y - y 0^2 = 4 r^2 , respectively. If z_0 = x_0 + i⁡ y_0…

Question

Let complex numbers α and 1/(α ¯) lie on circles x - x 0^2 + y - y 0^2 = r^2 , and x - x 0^2 + y - y 0^2 = 4 r^2 , respectively. If z_0 = x_0 + i⁡ y_0 satisfies the equation 2 z 0^2 = r⁡^2 + 2 , then α =

MARKS Content Team · Accepted Answer

Correct answer: (C) 1/(√7). z - z_0 = r z - z_0 = 2 r α - z_0 = r 1/(α¯)-z_0=2rαα¯=α^2 (α)/(α^2) - z_0 = 2 r α - z_0 α ¯ - z ¯_0 = r^2 ⇒ α^2 - z_0 α ¯ - α z ¯_0 + z 0^2 = r^2 (α)/(α^2) - z_0 (α ¯)/(α^2) - z ¯_0 = 4 r^2 ⇒ (α^2)/(α^4) - (z ¯_0 α)/(α^2) - (z ¯_0 α)/(α^2) + z 0^2 = 4 r^2 1 - z_0 α ¯ - z ¯_0 α + z 0^2 α^2 = 4 r^2 α^2 ⇒ α^2 - 1 + z 0^2 1 - α^2 = r^2 1 - 4 α^2 α^2 - 1 1 - (r^2 + 2)/2 = r^2 1 - 4 α^2 α^2 - 1 (- r^2)/2 = r^2 1 - 4 α^2 α^2 - 1 = - 2 + 8 α^2 1 = 7 α^2 ⇒ α = 1/(√7)

Let complex numbers α and $\frac{1}{\bar{α}}$ lie on circles ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2},$ and ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = {4 r}^{2},$ respectively. If $z_{0} = x_{0} + i y_{0}$ satisfies the equation $2 {|z_{0}|}^{2} = r^{2} + 2, then |α| =$

Solution

Relevant Questions

Let complex numbers α and 1 α ¯ lie on circles x - x 0 2 + y - y 0 2 = r 2 , and x - x 0 2 + y - y 0 2 = 4 r 2 , respectively. If z 0 = x 0 + i⁡ y 0 satisfies the equation 2 z 0 2 = r⁡ 2 + 2 , then α =

Solution

Relevant Questions

Let complex numbers α and $\frac{1}{\bar{α}}$ lie on circles ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2},$ and ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = {4 r}^{2},$ respectively. If $z_{0} = x_{0} + i y_{0}$ satisfies the equation $2 {|z_{0}|}^{2} = r^{2} + 2, then |α| =$