Which of the following statement(s) is(are) correct ?

Options

A. If the imaginary part of `(2z+1)/(iz+1)` is -2, then the locus of the point representing z in the complex plane is a straight line.
B. Two circles are given such that one is completely lying inside the other without touching. Then the locus of the centre of a variable circle which touches the smaller circle from outside and the bigger circle from inside in an ellipse.
C. If the imaginary part of `(2z+1)/(iz+1)` is -2, then the locus of the point representing z in the complex plane is a circle.
D. Two circles are given such that one is completely lying inside the other without touching. Then the locus of the centre of a variable circle which touches the smaller circle from outside and the bigger circle from inside is a hyperbola.

Answer

A, B

Explanation

(A) We have ` (2z+1)/(iz+1)=(2(x+iy)+1)/(i(x+iy)+1)=((2x+1)+2y)/((1-y)+ix)`
`=([(2x+1)(1-y)+2xy]+i[2y(1-y)-x(2x+1)])/((1-y^(2))+x^(2))`
But it is given that imaginary part of `((2z+1))/(iz+1))` is -2
`rArr x +2y-2=0` . Which is a straight line.
(B) In the figure, circles with hard lines are the given circles with centres `C_(1)` and `C_(2)` and radii `r_(1)` and `r_(2)`.
Let the circle with dotted line be the variable circle, which touches the given two circles as explained in the question, which has centre C and radius r.
Now `C C_(2)=r+r_(2)` and `C C_(1)=r_(1)-r`
Hence, `C C_(1)+C C_(2)=r_(1)+r_(2)` (=constant)
Then the locus of C is an ellipse whose foci are `C_(1)` and `C_(2)`