Home
Class 12
MATHS
For three non-colliner complex numbers Z...

For three non-colliner complex numbers `Z,Z_(1)` and `Z_(2)` prove that `|Z-(Z_(1)+Z_(2))/(2)|^(2) + |(Z_(1) -Z_(2))/(2)|^(2)= (1)/(2) | Z - Z_(1)|^(2) +(1)/(2)|Z- Z_(2)|^(2)`

Text Solution

Verified by Experts

Consider the formed by A(Z), `B(Z_(1))` and `C(Z_(2))`.
Let midpoint of BC be D having complex number `(Z_(1)+Z_(2))/(2)`
By Apollononius therorem, we have
`AB^(2) + AC^(2) + 2(AD^(2) + BD^(2))`
`therefore |Z-(Z_(1) +Z_(2))/(2)|^(2) + |(Z_(1) +Z_(2))/(2)|`
`AD^(2) + BD^(2)`
`=(1)/(2)AB^(2) + (1)/(2)Ac^(2)`
`= (1)/(2)|Z- Z_(1)|^(2)+(1)/(2)|Z-Z_(2)|^(2)`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.10|10 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.11|6 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.8|11 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE ENGLISH|Exercise All Questions|101 Videos

Similar Questions

Explore conceptually related problems

Prove that |z_(1) + z_(2)|^(2) + |z_(1)-z_(2)|^(2)=2|z_(1)|^(2) + 2|z_(2)|^(2)

For two complex numbers z_(1) and z_(2) , we have |(z_(1)-z_(2))/(1-barz_(1)z_(2))|=1 , then

If the complex number Z_(1) and Z_(2), arg (Z_(1))- arg(Z_(2)) =0 . then show that |z_(1)-z_(2)| = |z_(1)-z_(2)| .

If the complex number Z_(1) and Z_(2), arg (Z_(1))- arg(Z_(2)) =0 . then show that |z_(1)-z_(2)| = |z_(1)|-|z_(2)| .

If z_(1)=3 + 4i,z_(2)= 8-15i , verify that |z_(1) + z_(2)|^(2) + |z_(1)-z_(2)|^(2)= 2(|z_(1)|^(2) + |z_(2)|^(2))

For any two complex numbers z_1 and z_2 , prove that Re(z_1z_2)=Re(z_1) Re(z_2)-Im(z_1) Im(z_2) .

For any two complex numbers z_(1) and z_(2) |z_(1)+z_(2)|^(2) =(|z_(1)|^(2)+|z_(2)|^(2))

For any two complex numbers z_(1),z_(2) the values of |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2) , is

Consider the complex numbers z_(1) and z_(2) Satisfying the relation |z_(1)+z_(2)|^(2)=|z_(1)|^2 + |z_(2)|^(2) Complex number z_(1)/z_(2) is

Consider the complex numbers z_(1) and z_(2) Satisfying the relation |z_(1)+z_(2)|^(2)=|z_(1)| + |z_(2)|^(2) Complex number z_(1)//z_(2) is