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4-1.Complex numbers
medium
यदि ${z_1},{z_2}$ दो सम्मिश्र संख्याएँ इस प्रकार हैं कि ${z_1} + {z_2}$ तथा ${z_1}{z_2}$ दोनों वास्तविक हैं, तब
A
${z_1} = - {z_2}$
B
${z_1} = {\bar z_2}$
C
${z_1} = - {\bar z_2}$
D
${z_1} = {z_2}$
Solution
(b) माना ${z_1} = a + ib,{z_2} = c + id$, तब ${z_1} + {z_2}$ वास्तविक है
$⇒(a + c) + i(b + d)$ वास्तविक है
$⇒ b + d = 0$ $⇒ d = – b$ …..(i)
${z_1}{z_2}$ वास्तविक है $⇒ (ad – bd) + i(ac + bc)$ वास्तविक है
$⇒ ad + bc = 0$
$⇒ a( – b) + bc = 0 ⇒ a = c$
${z_1} = a + ib = c – id = {\bar z_2}$ $(\because a = c$ एवं $b = – d)$
Standard 11
Mathematics