Let two circles C1 and C2 be given the equation of C1 is|z2|+a¯z+az¯+c=0 and equation of C2 is |z2|+b¯z+bz¯+d=0 where a and b are complex number and c and d are real number, then

The equation x2+y2+2gx+2fy+c=0 can be reduced to |z|2+(g−if)z+(g+if)z¯+c=0 by putting z=x+iy⇒z¯=x−iyi) Radius of c1=|a|2−cii) If C1 and C2 touch extremely, then distance between centres =r1+r2⇒|a¯−b¯|=|a|2−c+|b|2−c iii) Let a=g1+if1; g2+if2c1c2 will cut orthogonally if2g1g2+2f1f2=c+d Now, ab¯+ba¯=(gn+if1)(g2−if2)+(g2+if2)(g1+if1)=2g1g2+2f1f2iv) Radical axis can be obtained by subtracting the equations

Let two circles C1 and C2 be given the equation of C1 is|z2|+a¯z+az¯+c=0 and equation of C2 is |z2|+b¯z+bz¯+d=0 where a and b are complex number and c and d are real number, then

The equation x2+y2+2gx+2fy+c=0 can be reduced to |z|2+(g−if)z+(g+if)z¯+c=0 by putting z=x+iy⇒z¯=x−iyi) Radius of c1=|a|2−cii) If C1 and C2 touch extremely, then distance between centres =r1+r2⇒|a¯−b¯|=|a|2−c+|b|2−c iii) Let a=g1+if1; g2+if2c1c2 will cut orthogonally if2g1g2+2f1f2=c+d Now, ab¯+ba¯=(gn+if1)(g2−if2)+(g2+if2)(g1+if1)=2g1g2+2f1f2iv) Radical axis can be obtained by subtracting the equations

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Let two circles $C_{1} a n d C_{2}$ be given the equation of $C_{1}$ is $| z^{2} | + \bar{a} z + a \bar{z} + c = 0$ and equation of $C_{2}$ is $| z^{2} | + \bar{b} z + b \bar{z} + d = 0$ where a and b are complex number and c and d are real number, then

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radius of $C_{1}$ is $\sqrt{{| a |}^{2} - c}$

$C_{1} a n d C_{2}$ will touch externally iff $| a - b | = \sqrt{{| a |}^{2} - c} + \sqrt{| b^{2} | - d}$

$C_{1} a n d C_{2}$ will intersect orthogonally iff $a \bar{b} + \bar{a} b = c + d$

radical axis of $C_{1} a n d C_{2}$ is $(\bar{a} - \bar{b}) z + (a - b) \bar{z} + c - d = 0$

answer is A, B, C, D.

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Detailed Solution

The equation $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ can be reduced to ${| z |}^{2} + (g - i f) z + (g + i f) \bar{z} + c = 0$ by putting $z = x + i y \Rightarrow \bar{z} = x - i y$
i) Radius of $c_{1} = \sqrt{{| a |}^{2} - c}$
ii) If $C_{1} a n d C_{2}$ touch extremely, then distance between centres $= r_{1} + r_{2}$
$\Rightarrow | \bar{a} - \bar{b} | = \sqrt{{| a |}^{2} - c} + \sqrt{{| b |}^{2} - c}$
iii) Let $a = g_{1} + i f_{1}; g_{2} + i f_{2}$ $c_{1} c_{2}$ will cut orthogonally if $2 g_{1} g_{2} + 2 f_{1} f_{2} = c + d$
Now, $a \bar{b} + b \bar{a} = (g_{n} + i f_{1}) (g_{2} - i f_{2}) + (g_{2} + i f_{2}) (g_{1} + i f_{1}) = 2 g_{1} g_{2} + 2 f_{1} f_{2}$
iv) Radical axis can be obtained by subtracting the equations