Given f ( z ) = z 2 − 19 z , there are complex plane z with the property that z , f ( z ) and f ( f ( z )) are the vertices of a right triangle in the complex plane with a right angle at f ( z ) . the

Question

Given f ( z ) = z 2 − 19 z , there are complex plane z with the property that z , f ( z ) and f ( f ( z )) are the vertices of a right triangle in the complex plane with a right angle at f ( z ) . there are positive integers m and n such that one such value of z is m + n ​ + 11 i , then

TestHub · Accepted Answer

f ( z ) z − f ( z ) f ( f ( z )) − f ( z ) ​ z − ( z 2 − 19 z ) ( z 2 − 19 z ) 2 − 19 ( z 2 − 19 z ) − ( z 2 − 19 z ) ​ L e t z T h ere f ore , a 2 − 18 a − 140 a ​ = z 2 = − 19 z i s ima g ina ry = ( 20 z − z 2 ) ( z 2 − 19 z ) ( z 2 − 19 z − 20 ) ​ = − z ( z − 20 ) z ( z − 19 ) ( z − 20 ) ( z + 1 ) ​ = − ( z − 19 ) ( z + 1 ) i s ima g ina ry = − ( z 2 − 18 z − 19 ) = a + 11 i = − (( a + 11 i ) 2 − 18 ( a + 11 i ) − 19 ) = − ( a 2 − 121 + 22 ai − 18 a − 198 i − 19 ) = − ( a 2 − 18 a − 140 + i ( 22 a − 198 )) i s ima g ina ry = 0 = 2 ( 1 ) 18 ± ( − 18 ) 2 − 4 ( 1 ) ( − 140 ) ​ ​ = 2 18 ± 324 + 560 ​ ​ = 2 18 ± 884 ​ ​ = 2 18 ± 2 221 ​ ​ = 9 ± 221 ​ ​

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JEE Advanced (2028)