Home
Class 12
MATHS
Let a, b, c be distinct complex numbers ...

Let `a`, `b`, `c` be distinct complex numbers with `|a|=|b|=|c|=1` and `z_(1)`, `z_(2)` be the roots of the equation `az^(2)+bz+c=0` with `|z_(1)|=1`. Let `P` and `Q` represent the complex numbers `z_(1)` and `z_(2)` in the Argand plane with `/_POQ=theta`, `o^(@) lt 180^(@)` (where `O` being the origin).Then

A

`b^(2)=ac`, `theta=(2pi)/(3)`

B

`theta=(2pi)/(3)`,`PQ=sqrt(3)`

C

`PQ=2sqrt(3)`, `b^(2)=ac`

D

`theta=(pi)/(3)`, `b^(2)=ac`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`(a,b)` `|z_(1)+z_(2)|=|-(b)/(a)|=1`
`|z_(1)z_(2)|=|(c )/(a)|=1`
`:.|z_(2)|=1`
`|z_(1)+z_(2)|^(2)=1`
`:.2+z_(1)barz_(2)+z_(2)barz_(1)=1`
Now `z_(2)=z_(1)e^(itheta)`
`:. |z_(1)+z_(1)e^(itheta)|=|z_(1)||1+e^(itheta)|=1`
`:.2cos"(theta)/(2)=1`
`:.theta=(2pi)/(3)`
Now, `((z_(1)+z_(2))^(2))/(z_(1)z_(2))=1`
`implies (b^(2))/(a^(2))=(c )/(a)`
`impliesb^(2)=ac`
`PQ=|z_(1)-z_(2)|=sqrt(3)`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE PUBLICATION|Exercise Matching Column|1 Videos

  • COMPLEX NUMBERS

    CENGAGE PUBLICATION|Exercise Comprehension|11 Videos

  • COMPLEX NUMBERS

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE PUBLICATION|Exercise All Questions|102 Videos

Similar Questions

Explore conceptually related problems

If one root of the equation z^2-a z+a-1= 0 is (1+i), where a is a complex number then find the root.

If z_(1)and z_(2) are conjugate complex number, then z_(1)+z_(2) will be

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 represents

Let p and q are complex numbers such that |p|+|q| lt 1 . If z_(1) and z_(2) are the roots of the z^(2)+pz+q=0 , then which one of the following is correct ?

let z_1 and z_2 be roots of the equation z^2+pz+q=0 where the coefficients p and q may be complex numbers let A and B represnts z_1 and z_2 in the complex plane if angleAOB=alpha ne 0 and OA=OB where 0 is the origin prove that p^2=4qcos^2(alpha/2)

Let z be a complex number satisfying |z+16|=4|z+1| . Then

If z ne 1 and (z^2)/(z-1) is real then the point represented by the complex number z lies

If zne1and(z^(2))/(z-1) is real , then the point represented by the complex number z lies -

Let z_(1) and z_(2) be the roots of z ^(2)+az+b=0 . If the origin, z_(1) and z_(2) form an equilateral triangle. Then-

A point P which represents a complex number z, moves such that |z-z_(1)|= |z-z_(2)|, then the locus of P is-