jee-advanced 2003 Q1

jee-advanced · India · mains Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based)
If $z _ { 1 }$ and $z _ { 2 }$ are two complex numbers such that $\left| \mathrm { z } _ { 1 } \right| < 1 < \left| \mathrm { z } _ { 1 } \right|$ then prove that $$\left| \frac { 1 - z _ { 1 } \bar { z } _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| < 1 .$$
The centre of the circle inscribed in the square determined by the two pairs of lines $x ^ { 2 } - 8 x + 12 = 0$ and $y ^ { 2 } - 14 y + 45 = 0$ is 
If $z _ { 1 }$ and $z _ { 2 }$ are two complex numbers such that $\left| \mathrm { z } _ { 1 } \right| < 1 < \left| \mathrm { z } _ { 1 } \right|$ then prove that $$\left| \frac { 1 - z _ { 1 } \bar { z } _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| < 1 .$$
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