Home
Class 12
MATHS
Let z1 \and\ z2 be the roots of the equa...

Let `z_1 \and\ z_2` be the roots of the equation `z^2+p z+q=0,` where the coefficients `p \and \q` may be complex numbers. Let `A \and\ B` represent `z_1 \and\ z_2` in the complex plane, respectively. If `/_A O B=theta!=0 \and \O A=O B ,\where\ O` is the origin, prove that `p^2=4q"cos"^2(theta//2)dot`

Text Solution

Verified by Experts

Let `z_(1)` and `z_(2)` be roots of the equation `z^(2)+ pz + q =0` .
Then,
`z_(1) +z_(2) = -p,z_(1)z_(2) =q`
Also, `(z_(2))/(z_(1)) = e^(itheta) or z_(2)= z_(1)e^(itheta) `
`rArr z_(1) (1+ e^(itheta)) = -p,z_(1)^(2) e^(itheta) = q`
`z_(1)^(2) = q e^(itheta) = (p^(2))/((1+e^(itheta))^(2))`
`rArr p^(2) = qe^(-itheta) (1+e^(2itheta) + 2e^(itheta))`
`= q(e^(-itheta) + e^(itheta) + 2)`
`=q(2cos theta + 2)`
`= 4q cos ^(2) .(theta)/(2)`
Promotional Banner

Topper's Solved these Questions

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.1|4 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise EXERCISE3.2|9 Videos

  • COMPLEX NUMBERS

    CENGAGE ENGLISH|Exercise ILLUSTRATION|110 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

  • CONIC SECTIONS

    CENGAGE ENGLISH|Exercise All Questions|101 Videos

Similar Questions

Explore conceptually related problems

z_(1) and z_(2) are the roots of the equaiton z^(2) -az + b=0 where |z_(1)|=|z_(2)|=1 and a,b are nonzero complex numbers, then

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.

Solve the equation z^2 +|z|=0 , where z is a complex number.

Let z_1 and z_2 be theroots of the equation z^2+az+b=0 z being compex. Further, assume that the origin z_1 and z_2 form an equilatrasl triangle then

Roots of the equation x^2 + bx + 45 = 0 , b in R lie on the curve |z + 1| = 2sqrt(10), where z is a complex number then

Let z_(1) and z_(2) be the roots of the equation z^(2)+pz+q=0 . Suppose z_(1) and z_(2) are represented by points A and B in the Argand plane such that angleAOB=alpha , where O is the origin. Statement-1: If OA=OB, then p^(2)=4q cos^(2)alpha/2 Statement-2: If affix of a point P in the Argand plane is z, then ze^(ia) is represented by a point Q such that anglePOQ =alpha and OP=OQ .

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

Let z_1, z_2 be two complex numbers represented by points on the circle |z_1|= and and |z_2|=2 are then

If A and B represent the complex numbers z_(1) and z_(2) such that |z_(1)+z_(2)|=|z_(1)-z_(2)| , then the circumcenter of triangleOAB , where O is the origin, is

Let OP.OQ=1 and let O,P and Q be three collinear points. If O and Q represent the complex numbers of origin and z respectively, then P represents