Let `z_1a n dz_2` b be toots of the equation `z^2+p z+q=0,` where he coefficients `pa n dq` may be complex numbers. Let `Aa n dB` represent `z_1a n dz_2` in the complex plane, respectively. If `/_A O B=theta!=0a n dO A=O B ,w h e r eO` is the origin, prove that `p^2=4q"cos"^2(theta//2)dot`

Explanation

Since `z_1 +z_2 =- and z_1z_2 = q`
Now `z_1/z_2=|z_1|/|z_2|(cos alpha + I sin alpha )`
`rArr z_1/z_2 =(cos alpha + I sin alpha )/(1)`
`[ because |z_1| = |z_2|]`
Applying componendo and dividendo, we get
`(z_1+z_2)/(z_1-z_2)=(cosalpha+ i sin alpha + 1)/(cos alpha + i sin alpha-1)`
`=(2 cos^2(alpha//2)+2 i sin(alpha//2)cos(alpha//2))/(-2 sin^2(alpha//2)+2 isin(alpha//2)cos (alpha//2))`
`=(2 cos (alpha//2)[cos(alpha//2)+ i sin (alpha//2)])/(2i sin(alpha//2)[cos (alpha//2)+sin(alpha//2)])`
`=(cot (alpha //2))/(i)= - i cot alpha//2 rArr (-p)/(z_1 - z_2)=-i cot (alpha //2)`
On squaring both sides , we get `p^2/(z_1-z_2)^2=-i cot (alpha//2)`
`rArr " " p^2/((z_1+z_2)^2-4z_1z_2)=- cot^2 (alpha//2)`
`rArr p^2/(p^2 -4q)=- cot ^2(alpha//2)`
`rArr p^2= - p^2cot^2(alpha//2)+4q cot^2(alpha//2)`
`rArr p^2(1+cot^2 alpha//2)=4q cot^2 (alpha//2)`
`p^2 cosec^2 (alpha //2)= 4q cot^2 (alpha// 2)`
`rArr p^2 = 4q cos^2 alpha//2`