Prove that `|Z-Z_1|^2+|Z-Z_2|^2=a`will represent a real circle [with center `(|Z_1+Z_2|^//2+)`] on the Argand plane if `2ageq|Z_1-Z_1|^2`

Prove that `|Z-Z_1|^2+|Z-Z_2|^2=a`will represent a real circle [with c

1.	Prove that `\|Z-Z_1\|^2+\|Z-Z_2\|^2=a`will represent a real circle [with center `(\|Z_1+Z_2\|^//2+)`] on the Argand plane if `2ageq\|Z_1-Z_1\|^2`
Answer» `\|z-z_(1)\|^(2)+\|z-z_(2)\|^(2)=a` `implies(z-z_(1))(bar(z)-bar(z)_(1))+(z-z_(2))(bar(z)-bar(z)_(2))=a` `implies2zbar(z)-z(bar(z)_(1)+bar(z)_(2))-bar(z)(z_(1)+z_(2))+z_(1)bar(z)_(1)+z_(2)bar(z)_(2)=a` `implieszbar(z)=z((bar(z_(1))+bar(z_(2)))/(2))-bar(z)((z_(1)+z_(2))/(2))+(z_(1)bar(z)_(1)+z_(2)bar(z)_(2)-a)/(2)=0" "(1)` Above equation is of the form `zbar(z)+alphabar(z)+bar(alpha)z+beta=0`, which represents the real circle if `alphabar(alpha)-betage0` `implies((z_(1)+z_(2))(bar(z_(1))+bar(z_(2))))/(4)ge(z_(1)bar(z_(1))+z_(2)bar(z_(2))-a)/(2)` `implies2agez_(1)bar(z_(1))+z_(2)bar(z_(2))-z_(1)bar(z_(2))-z_(2)bar(z_(1))` `implies2age\|z_(1)-z_(2)\|^(2)`

Discussion

No Comment Found

Related InterviewSolutions

Prove that the distance of the roots of the equation `|sintheta_1|z^3+|sintheta_2|z^2+|sintheta_3|z+|sintheta_4|=3fromz=0`is greater than `2//3.`
` theta in [0,2pi]` and `z_(1)`, `z_(2)`, `z_(3)` are three complex numbers such that they are collinear and `(1+|sin theta|)z_(1)+(|cos theta|-1)z_(2)-sqrt(2)z_(3)=0`. If at least one of the complex numbers `z_(1)`, `z_(2)`, `z_(3)` is nonzero, then number of possible values of `theta` isA. InfiniteB. `4`C. `2`D. `8`
Find the locus of point `z`if `z ,i ,a n di z ,`are collinear.
Let `O`, `A`, `B` be three collinear points such that `OA.OB=1`. If `O` and `B` represent the complex numbers `O` and `z`, then `A` representsA. `(1)/(barz)`B. `(1)/(z)`C. `barz`D. `z^(2)`
If `A(z_(1))`, `B(z_(2))`, `C(z_(3))` are vertices of a triangle such that `z_(3)=(z_(2)-iz_(1))/(1-i)` and `|z_(1)|=3`, `|z_(2)|=4` and `|z_(2)+iz_(1)|=|z_(1)|+|z_(2)|`, then area of triangle `ABC` isA. `(5)/(2)`B. `0`C. `(25)/(2)`D. `(25)/(4)`
If the imaginery part of `(z-3)/(e^(itheta))+(e^(itheta))/(z-3)` is zero, then `z` can lie onA. a circle with unit radiusB. a circle with radius `3` unitsC. a straight line through the point `(3,0)`D. a parabola with the vertex `(3,0)`
Consider the region `R` in the Argand plane described by the complex number. `Z` satisfying the inequalities `|Z-2| le |Z-4|`, `|Z-3| le |Z+3|`, `|Z-i| le |Z-3i|`, `|Z+i| le |Z+3i|` Answer the followin questions : Minimum of `|Z_(1)-Z_(2)|` given that `Z_(1)`, `Z_(2)` are any two complex numbers lying in the region `R` isA. `0`B. `5`C. `sqrt(13)`D. `3`
Consider the region `R` in the Argand plane described by the complex number. `Z` satisfying the inequalities `|Z-2| le |Z-4|`, `|Z-3| le |Z+3|`, `|Z-i| le |Z-3i|`, `|Z+i| le |Z+3i|` Answer the followin questions : The maximum value of `|Z|` for any `Z` in `R` isA. `5`B. `14`C. `sqrt(13)`D. `12`
The roots of the equation `z^(4) + az^(3) + (12 + 9i)z^(2) + bz = 0` (where a and b are complex numbers) are the vertices of a square. Then The value of `|a-b|` isA. `5sqrt(5)`B. `sqrt(130)`C. 12D. `sqrt(175)`
If `z = (3+ 7i) (a +ib)` where a,`b in in Z- {0}`, is purely imaginary , then the minimum value of `|z|` isA. 74B. 45C. 58D. 65

Prove that `|Z-Z_1|^2+|Z-Z_2|^2=a`will represent a real circle [with center `(|Z_1+Z_2|^//2+)`] on the Argand plane if `2ageq|Z_1-Z_1|^2`

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment