The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle $${x^2} + {y^2} = 9$$is

$${\rm{z }} \ne {\rm{0}}$$ is a complex number

Column I

(A) Re z = 0
(B) Arg $$z = {\pi \over 4}$$

Column II

(p) Re$${z^2}$$ = 0
(q) Im$${z^2}$$ = 0
(r) Re$${z^2}$$ = Im$${z^2}$$

Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3.

Let $$\alpha \,,\,\beta $$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation $$(x - \alpha \,)\,(x - \beta ) + c = 0$$ are