QJEEM21Mab581: JEE Main 2021 Mathematics - Class 11 JEEM21Mab581 Solution: A point z moves in the complex plane such that , then the…

Step-by-Step Verified Solution

Let $z = x + iy$

$\arg \left( {{{x - 2 + iy} \over {x + 2 + iy}}} \right) = {\pi \over 4}$

$\arg (x - 2 + iy) - \arg (x + 2 + iy) = {\pi \over 4}$

${\tan ^{ - 1}}\left( {{y \over {x - 2}}} \right) - {\tan ^{ - 1}}\left( {{y \over {x + 2}}} \right) = {\pi \over 4}$

${{{y \over {x - 2}} - {y \over {x + 2}}} \over {1 + \left( {{y \over {x - 2}}} \right).\left( {{y \over {x + 2}}} \right)}} = \tan {\pi \over 4} = 1$

${{xy + 2y - xy + 2y} \over {{x^2} - 4 + {y^2}}} = 1$

$4y = {x^2} - 4 + {y^2}$

${x^2} + {y^2} - 4y - 4 = 0$

locus is a circle with center (0, 2) & radius = $2\sqrt 2$



min. value = ${(AP)^2} = {(OP - OA)^2}$

$= {\left( {9\sqrt 2 - 2\sqrt 2 } \right)^2}$

$= {\left( {7\sqrt 2 } \right)^2} = 98$