Question

find the locus of Z if |3z-5|=3|z+1| where z=x+iy​

Answer 1

Answer:  The required locus of the given point is $y^2+6x-2=0.$

Step-by-step explanation:  We are given to find the locus of z if |3z-5|=3|z+1| where z = x + i y​.

We know that for a complex number a + ib, we have

$|a+ib|=\sqrt{a^2+b^2}.$

According to the question, we have

$|3z-5|=3|z+1|\\\\\Rightarrow |3(x+iy)-5|=3|x+iy+1|\\\\\Rightarrow |(3x-5)+iy|=3|(x+1)+iy|\\\\\Rightarrow \sqrt{(3x-5)^2+y^2}=\sqrt{3^2((x+1)^2+y^2)}\\\\\Rightarrow 9x^2-30x+25+y^2=9(x^2+2x+1+y^2)~~~~~~~~~~~~~~~~~~~~[\textup{Squaring both sides}]\\\\\Rightarrow 9x^2+y^2-30x+25=9x^2+9y^2+18x+9\\\\\Rightarrow 9y^2-y^2+18x+30x+9-25=0\\\\\Rightarrow 8y^2+48x-16=0\\\\\Rightarrow y^2+6x-2=0.$

Thus, the required locus of the given point is $y^2+6x-2=0.$