Intercept made by the circle ` zbar z +bar a+abar z+r=0` on the real axis on complex plane is a.`sqrt((a+ bar a )-r)` b. `sqrt((a+ bar a )^2-r)` c. `sqrt((a+ bar a )^2-4r)` d. `sqrt((a+ bar a )^2-4r)`

Identify the locus of z if bar z = bar a +(r^2)/(z-a).

Find the radius and centre of the circle z bar(z) - (2 + 3i)z - (2 - 3i)bar(z) + 9 = 0 where z is a complex number

Prove that the circles z bar z +z( bar a )_1+bar z( a )_1+b_1=0 ,b_1 in R and z bar z +z( bar a )_2+ bar z a_2+b_2=0, b_2 in R will intersect orthogonally if 2R e(a_1( bar a )_2)=b_1+b_2dot

Let a ,b ,a n dc be any three nonzero complex number. If |z|=1a n d' z ' satisfies the equation a z^2+b z+c=0, prove that a ( bara) =c (barc) a n d|a||b|=sqrt(a c( bar b )^2)

Prove that the locus of midpoint of line segment intercepted between real and imaginary axes by the line bara z + a bar z+b=0,w h e r eb is a real parameterand a is a fixed complex number with nondzero real and imaginary parts, is a z+ bara barz =0.

For any complex number z prove that |R e(z)|+|I m(z)|<=sqrt(2)|z|

If z is non zero complex number, such that 2i z^(2) = bar(z), then |z| is

If z=z_0+A( bar z -( bar z _0)), w h e r eA is a constant, then prove that locus of z is a straight line.

The largest area of the trapezium inscribed in a semi-circle or radius R , if the lower base is on the diameter, is (a) (3sqrt(3))/4R^2 (b) (sqrt(3))/2R^2 (c) (3sqrt(3))/8R^2 (d) R^2