Greatest value of $|z|$ such that $|z-2i|le2$ and $ 0le arg(z+2)le 45^circ$












1














I've to sketch the complex number $z$ such that it satisfy both the inequality
$|(z-2i)|le2$ and $ 0le arg(z+2)le 45^circ$



I was able to sketch and shade the region that satisfies both inequality, Please see the below link.



Sketch on Argand diagram



I've a problem in getting the greatest value of $|z|$ (maximum length of $z$ from $(0,0),$ it should come around 3.70)






complex-numbers







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  • looks ok to me. what is your problem with $|z|$?
    – David Holden
    Oct 11 '14 at 3:28










  • The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
    – Platehead
    Oct 11 '14 at 3:37










  • I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
    – Arif
    Oct 11 '14 at 4:26
















1














I've to sketch the complex number $z$ such that it satisfy both the inequality
$|(z-2i)|le2$ and $ 0le arg(z+2)le 45^circ$



I was able to sketch and shade the region that satisfies both inequality, Please see the below link.



Sketch on Argand diagram



I've a problem in getting the greatest value of $|z|$ (maximum length of $z$ from $(0,0),$ it should come around 3.70)






complex-numbers







share|cite|improve this question
























  • looks ok to me. what is your problem with $|z|$?
    – David Holden
    Oct 11 '14 at 3:28










  • The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
    – Platehead
    Oct 11 '14 at 3:37










  • I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
    – Arif
    Oct 11 '14 at 4:26














1












1








1


1





I've to sketch the complex number $z$ such that it satisfy both the inequality
$|(z-2i)|le2$ and $ 0le arg(z+2)le 45^circ$



I was able to sketch and shade the region that satisfies both inequality, Please see the below link.



Sketch on Argand diagram



I've a problem in getting the greatest value of $|z|$ (maximum length of $z$ from $(0,0),$ it should come around 3.70)






complex-numbers







share|cite|improve this question















I've to sketch the complex number $z$ such that it satisfy both the inequality
$|(z-2i)|le2$ and $ 0le arg(z+2)le 45^circ$



I was able to sketch and shade the region that satisfies both inequality, Please see the below link.



Sketch on Argand diagram



I've a problem in getting the greatest value of $|z|$ (maximum length of $z$ from $(0,0),$ it should come around 3.70)






complex-numbers




complex-numbers






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Sep 18 at 13:14









Michael Hardy

1




1










asked Oct 11 '14 at 3:22









Arif

1261514




1261514












  • looks ok to me. what is your problem with $|z|$?
    – David Holden
    Oct 11 '14 at 3:28










  • The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
    – Platehead
    Oct 11 '14 at 3:37










  • I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
    – Arif
    Oct 11 '14 at 4:26


















  • looks ok to me. what is your problem with $|z|$?
    – David Holden
    Oct 11 '14 at 3:28










  • The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
    – Platehead
    Oct 11 '14 at 3:37










  • I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
    – Arif
    Oct 11 '14 at 4:26
















looks ok to me. what is your problem with $|z|$?
– David Holden
Oct 11 '14 at 3:28




looks ok to me. what is your problem with $|z|$?
– David Holden
Oct 11 '14 at 3:28












The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
– Platehead
Oct 11 '14 at 3:37




The place you've marked an x on seems good, call it $z$. As you indicated in the diagram, $z$ is the 45 degree point on the radius 2 circle, which should be $(sqrt{2},sqrt{2})$, but we've shifted it up by $2$, so $z = sqrt{2} + (2+sqrt{2})i$. Alternatively, writing $z = x+iy$, we have $y = x + 2$ as well as $y = sqrt{4-x^2}+2$.
– Platehead
Oct 11 '14 at 3:37












I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
– Arif
Oct 11 '14 at 4:26




I want to find greatest length(its distance from (0,0) ) of $|z|$ it should be around 3.70 units, unable to understand how to work it out.
– Arif
Oct 11 '14 at 4:26










3 Answers
3






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oldest

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0














if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:



$$
frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
$$






share|cite|improve this answer





























    0














    Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
    $$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$






    share|cite|improve this answer





























      0














      Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.



      From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.



      Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.



      $therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$






      share|cite|improve this answer





















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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        0














        if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:



        $$
        frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
        $$






        share|cite|improve this answer


























          0














          if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:



          $$
          frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
          $$






          share|cite|improve this answer
























            0












            0








            0






            if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:



            $$
            frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
            $$






            share|cite|improve this answer












            if you parametrize the circle of constraint it is $(2cos t,2sin t +2)$ with $t=0$ corresponding to the point $(2,2)$. so:



            $$
            frac18 |z|^2 = frac12 (cos^2 t +sin^2 t + 2 sin t +1 )= 1+ sin t
            $$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Oct 11 '14 at 3:41









            David Holden

            14.6k21224




            14.6k21224























                0














                Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
                $$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$






                share|cite|improve this answer


























                  0














                  Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
                  $$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$






                  share|cite|improve this answer
























                    0












                    0








                    0






                    Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
                    $$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$






                    share|cite|improve this answer












                    Inspection of the figure indicates that the point $z_0:=sqrt{2}+i(2+sqrt{2})$ is the feasible point with largest absolute value: The closed disk $D_0$ with center $0$ and radius $|z_0|$ obviously contains the complete feasible region $F$, and $z_0in F$. It follows that
                    $$max_{zin F}|z|=|z_0|=sqrt{8+4sqrt{2}}doteq3.6955 .$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 14 '16 at 11:09









                    Christian Blatter

                    172k7112325




                    172k7112325























                        0














                        Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.



                        From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.



                        Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.



                        $therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$






                        share|cite|improve this answer


























                          0














                          Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.



                          From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.



                          Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.



                          $therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$






                          share|cite|improve this answer
























                            0












                            0








                            0






                            Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.



                            From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.



                            Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.



                            $therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$






                            share|cite|improve this answer












                            Let $0+2i$ be C (the centre of the circle), let the point of furthest intersection (greatest $|z|$) of the 2 loci be A (You have indicated it on your diagram with a cross), construct the point B such that $triangle CAB$ has a right angle at B.



                            From here we can note that $angle CAB=frac{pi}{4}$(given - construction of CB is such that it is parallel to the x-axis, use corresponding angles), $AC=2$ (radii). Using trigonometry we can find that $BC=2cos(frac{pi}{4})=sqrt{2}$, similarly $AB=2sin(frac{pi}{4})=sqrt{2}$.



                            Applying our knowledge to the diagram (the Argand diagram), we see that the real part of the desired complex number is given by $BC$, $Re(z)=sqrt{2}$. The imaginary part is the sum of $AB$ and the height (distance) from the number $sqrt{2}$ to $sqrt{2}+2i$ (distance is $2$), thus $Im(z)=BC+2=2+sqrt{2}$.



                            $therefore$ $z=sqrt{2}+i(2+sqrt{2})$ and $|z|=2sqrt{2+sqrt{2}}$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Apr 3 '17 at 19:59









                            frog1944

                            1,28711322




                            1,28711322






























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