1.

Consider the equationof line abarz + baraz + b=0, whereb is arealparameterand a isfixed non-zero complex number. The locus of mid-point of thelineintercepted between real and imaginary axis is givenby

Answer»

`az- BAR(az) =0`
`az + bar(az) =0`
`az-bar(az) + B =0`
`az - bar(az) + 2B = 0`

Solution :Givenequationof line is `abarz + abarz + b =0AA b in R`.
Let thePQbe thesegement intercept between axes.
Forintercept on real axis `Z_(R)`.
`Z =barz`
`rArr Z_(R)(a+ bara) + b =0`
` rArr Z_(R) = (-b)/(a + bara)`
For interceptonimaginary `Z_(1)`
`z +barz = 0`
`rArr Z_(1)(bara - a) + b=0`
`rArr Z_(1)= (b)/(a+bara)`
For mid-point,
`z= (Z_(R) + Z_(I))/(2)`
`rArr z =(-b)/(2)[(1)/(bara+a)+(1)/(bara +a)]`
`z= (barab)/((a + bara)(a-bara))`
`rArr z = (barab)/(a^(2) -(a)^(2))`
`(z[a^(2)-(a)^(2)])/(bara) = barz((bara)^(2) - (a)^(2))/(a)`
`rArr az + bar(az) =0`


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