4) If x, y are real and y 2 x - 5 i a n d 4 + i x + y 2 are conjugate to each other, find x and y. Share with your friends Share 0 Neha Sethi answered this Dear student Let z1=y2x-5i and z2=4+ix+y2⇒z1=y2x+5i and z2¯=4-ix+y2Given: z1=z2 and z2=z1⇒y2x+5i=4+ix+y2 and 4-ix+y2=y2x-5iConsider, y2x+5i=4+ix+y2Comparing real and imaginary parts, we gety2x=4 ...2 and 5=x+y2⇒y2=5-x ...1Putting 1 in 2, we get5-xx=4⇒5x-x2=4⇒-x2+5x-4=0⇒x2-5x+4=0⇒x2-4x-x+4=0⇒xx-4-1x-4=0⇒x-1x-4=0⇒x=1 or x=4Case I:When x=1, y2=5-1=4⇒y=±2So, we get 1,2 and 1,-2Case II: when x=4y2=5-4=1y=±1So, we get 4,1 and 4,-1Now consider, 4-ix+y2=y2x-5iComparing real and imaginary parts, we get4=y2x and x+y2=5 ....4⇒4x=y2 ...3Putting 3 in 4, we getx+4x=5⇒x2+4=5x⇒x2-5x+4=0⇒x2-4x-x+4=0⇒x2-4x-x+4=0⇒xx-4-1x-4=0⇒x-1x-4=0⇒x=1 or x=4 CAse I: when x=1Putting in 3, we get ⇒y2=4⇒y=±2So, we get 1,2 and 1,-2Case II: when x=4y2=±1⇒y=±1So, we get 4,1 and 4,-1So, we get 1,2 , 1,-2 , 4,1 and 4,-1 Regards 0 View Full Answer