Let 2 z ˉ + i z ˉ − i = 3 1 , z ∈ C , be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points ( 0 , 0 ) , C and ( α , 0 ) is 11 square unit

Question

Let ​ 2 z ˉ + i z ˉ − i ​ ​ = 3 1 ​ , z ∈ C , be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points ( 0 , 0 ) , C and ( α , 0 ) is 11 square units, then α 2 equals:

TestHub · Accepted Answer

Let z = x + i y ⇒ z ˉ = x − i y ​ 3∣ z ˉ − i ∣ = 1∣2 z ˉ + i ∣ = 3 ∣ ( x − ( y + 1 ) i ∣ = ∣2 x + i ( 1 − 2 y ) ∣ = 3 x 2 + ( y + 1 ) 2 ​ = ( 2 x ) 2 + ( 1 − 2 y ) 2 ​ = 9 ( x 2 + y 2 + 2 y + 1 ) = 4 x 2 + 4 y 2 − 4 y + 1 ⇒ 5 x 2 + 5 y 2 + 22 y + 8 = 0 ⇒ Centre ≡ ( 0 , − 5 11 ​ ) ​ Area of Δ = 2 1 ​ ∣ α ∣ ​ 5 − 11 ​ ​ = 11 ​ ⇒ ∣ α ∣ = 10 ⇒ α 2 = 100 ​

Mathematics - Complex Number Question with Solution | TestHub

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