QJEEM21Ma9b9a: JEE Main 2021 Mathematics - Class 11 JEEM21Ma9b9a Solution: If the least and the largest real values of a, for which the…

Question

Verified Solution • 2025–26 Session moderate

If the least and the largest real values of a, for which the
equation z +

\alpha

|z – 1| + 2i = 0 (z

\in

C and i =

\sqrt { - 1}

) has a solution, are p and q respectively; then 4(p² + q²) is equal to __________.

Accepted Answer

x + iy + \alpha \sqrt {{{(x - 1)}^2} + {y^2}} + 2i = 0

\therefore

y + 2 = 0 and

x + \alpha \sqrt {{{(x - 1)}^2} + {y^2}} = 0

y =

-

2 &

{x^2} = {\alpha ^2}({x^2} - 2x + 1 + 4)

{\alpha ^2} = {{{x^2}} \over {{x^2} - 2x + 5}} \Rightarrow {x^2}({\alpha ^2} - 1) - 2x{\alpha ^2} + 5{\alpha ^2} = 0

x \in R \Rightarrow D \ge 0

4{\alpha ^4} - 4({\alpha ^2} - 1)5{\alpha ^2} \ge 0

{\alpha ^2}[4{\alpha ^2} - 2{\alpha ^2} + 20] \ge 0

{\alpha ^2}[ - 16{\alpha ^2} + 20] \ge 0

{\alpha ^2}\left[ {{\alpha ^2} - {5 \over 4}} \right] \le 0

0 \le {\alpha ^2} \le {5 \over 4}

\therefore

{\alpha ^2} \in \left[ {0,{5 \over 4}} \right]

\therefore

\alpha \in \left[ { - {{\sqrt 5 } \over 2},{{\sqrt 5 } \over 2}} \right]

then

4[{(q)^2} + {(p)^2}] = 4\left[ {{5 \over 4} + {5 \over 4}} \right] = 10

Step-by-Step Verified Solution