Question

«please solve this question in detail»
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Answer 1

Answer:

Step-by-step explanation:

Given complex numbers are

$\tt{z_{1}=-3+iyx^2\,\,\,\,\&\,\,\,\,z_{2}=x^2+y+4i}$

According to the given condition,

$\sf{\bar{z_{1}}=z_{2}}$

$\sf{\implies\,-3-iyx^2=x^2+y+4i}$

Comparing the real and imaginary parts,

$\sf{\implies\,-3=x^2+y\,\,\,\,\&\,\,\,\,-yx^2=4}$

$\sf{\implies\,x^2=-\dfrac{4}{y}}$

So,

$\sf{\implies\,-3=-\dfrac{4}{y}+y}$

$\sf{\implies\,3=\dfrac{4}{y}-y}$

$\sf{\implies\,3y=4-y^2}$

$\sf{\implies\,y^2+3y-4=0}$

$\sf{\implies\,y^2+4y-y-4=0}$

$\sf{\implies\,y(y+4)-1(y+4)=0}$

$\sf{\implies\,(y-1)(y+4)=0}$

$\sf{\implies\,y=1\,\,\,\,\,or\,\,\,\,\,y=-4}$

Then,

$\sf{\implies\,x^2=-4\,\,\,\,\,or\,\,\,\,\,x^2=1}$

$\sf{\implies\,x=\pm2i\,\,\,\,\,or\,\,\,\,\,x=\pm1}$

So, real values of x and y are

$\sf{x=\pm1\,\,\,\,\,\,and\,\,\,\,\,\,y=-4}$