The locus of z=x+ iy, such that Im ⁡ ( z-3i iz+4 )=0 is

Question

Accepted Answer

Correct answer: C. (c) Given, z=x+ iy and Im ⁡ ( z-3i iz+4 )=0 ⇒ Im ⁡ ( x+ iy-3ii (x+ iy )+4 )=0 ⇒ Im ⁡ ( x+i (y-3 ) (4-y )+ ix )=0 ⇒ Im ⁡( x+i (y-3 ) (4-y )+ ix (4-y )- ix (4-y )- ix)=0 ⇒ Im ⁡ ( x (4-y )-ix^2+i (y-3 ) (4-y )+x (y-3 ) (4-y )^2+x^2 )=0 ⇒ Im ( x-i(x^2+y^2-7y+12) (4-y )^2+x^2 )=0 ⇒ (x^2+y^2-7y+12) (4-y )^2+x^2=0( ∵ (4-y )^2+x^2 0) ⇒4-y 0 andx 0 ⇒y 4 andx 0 ⇒x_2+y_2-7y+12=0 but (x ,y ) (0 ,4 ) .

$The locus of z = x + iy, such that Im (\frac{z - 3 i}{iz + 4}) = 0 is$

Solution

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