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Further Maths Argand Diagrams

Any ideas on 4b? Thank you!

IMG_1016.jpg427.3KB

Part a) must be a circle, radius 5, centered on (6,1).
Part b) Translates the origin to (11,10), then you have to find the smallest angle (most minimum) of a point on the circle from the new origin.
Can you sketch it?

Reply 2

RDKGames

Study Forum Helper

Original post by IzzyLJB

Any ideas on 4b? Thank you!

\arg(z-(11+10i))

is a half-line starting from

z_1 = 11+10i

going off in some direction (blue line in my diagrams below)

The direction is limited though, because

z

must be on the circle that is defined by

|z-6-i| = 5

(you sketch this circle in part a, it is the red one in my diagram).

Hence you are seeking for the minimum argument of the blue half-line (angle measured from the black dotted line in my diagram, labeled

\theta

) such that there is some intersection with the circle.

You need to look at two extremes. One of which is the following:

Spoiler

or this one:

Spoiler

Which one do you think represents the MINIMUM argument?? Work with that diagram and derive the answer for

\theta

(edited 5 years ago)

Reply 3

Mad Man

Can anyone help me with part b ? I'm guessing the minimum arguement is pi/2 right? So where do I go from here?

Reply 4

Mad Man

I think I've got it please correct me if I'm wrong, or if I need to be more elaborate. See the attactched file.

scan_jlambert_2020-10-30-20-16-30.pdf302.4KB

Looks good.

Original post by mqb2766

Looks good.

Is it worthy of full marks?

Reply 7

Mad Man

Pardon me, I forgot to put that sqrt(9^2 + 5^2)=sqrt(106)

Reply 8

mqb2766

Original post by Mad Man

Is it worthy of full marks?

It's a bit messy, but has the main points in. Hard for me to judge if you may be docked a mark.
For instance, you could describe why the two triangles are congruent, why the min angle corresponds to the tangent ... if you want to be guaranteed full marks. However, I don't have the mark scheme.

(edited 3 years ago)

Reply 9

Mad Man

Original post by mqb2766

It's a bit messy, but has the main points in. Hard for me to judge of you may be docked a mark.
For instance, you could describe why the two triangles are congruent, why the min angle corresponds to the tangent ... if you want to be guaranteed full marks. However, I don't have the mark scheme.