Complex numbers!

[Splash-Tackle-Flail]

^re: that, I'd highly recommend you read or go-over the "Loci" chapter in your GMA book. If you can "read" the graph in English as brightsky has done, and know what it is before doing any maths, you'll be steps ahead of the game

So would we be allowed to solve the question with that method (as in is loci in the spesh syllabus?)

[pinklemonade]

Just wanted to introduce a rather nifty method to determine the Cartesian equation of an ellipse from the 'complex equation'.

First, consider how you would draw an ellipse at a beach if all you were given were two pegs, a rope and a stick. To begin, you would hammer the two pegs into the sand, making sure that they were more or less side by side at a fixed distance from each other. Then, you would tie the rope around the two pegs, grab the stick, drag the rope that is now secured at both ends to the two pegs upwards until the rope becomes taut, and move the stick around the two pegs in a circular fashion, ensuring the rope remains taut.

Now, let F_1 (-m,0) and F_2 (m,0) denote the positions of the two pegs. In the diagram below, the dotted line represents the rope, pulled taut by the stick, which is positioned at P (x,y).

(Image removed from quote.)

Now, to obtain the full ellipse, we would take the stick positioned at P(x,y) and move it around F_1 and F_2 in a circular fashion.

(Image removed from quote.)

To understand the method of constructing an ellipse outlined above, acquaintance with the formal definition of an ellipse is required. Analytically, an ellipse may be defined as the set of all points P(x,y) such that PF_1 + PF_2 = k, where k is some constant. Geometrically, the sum of PF_1 and PF_2 represents the length of the string used above to construct the ellipse, and since the length of the string is fixed, it makes sense that the sum of PF_1 and PF_2 is a constant.

Now, recall the general equation of an ellipse x^2/a^2 + y^2/b^2 = 1. How do we get from the analytic definition of the ellipse provided above to the Cartesian equation? First, drag the stick, positioned at P(x,y), to the far right the ellipse. It is clear that the length of the string is 2a. We can thus conclude that k = 2a. Now, drag the stick to the top of the ellipse. By Pythagoras' theorem, m^2 + b^2 = (k/2)^2. Since k = 2a, this means that m^2 + b^2 = a^2.

Let us now turn our attention to the question that you posed above and try to apply the foregoing theory into practice. The 'complex equation' that we are given is:

|z - 1| + |z + 1| = 3

Read the equation above out loud in English: "the distance from z to 1 plus the distance from z to -1 is equal to 3". Sounds like an ellipse to me! Now, k = 3 in this case. Since we have already established that k = 2a, it follows that 2a = 3, which means a = 3/2. Now, m = 1. By Pythagoras' theorem, m^2 + b^2 = a^2. Plugging in the values of a and m yields the following equation:

1^2 + b^2 = (3/2)^2
1 + b^2 = 9/4
b^2 = 5/4
b = sqrt(5)/2

So, what is the Cartesian equation of the ellipse? x^2/(3/2)^2 + y^2/(sqrt(5)/2)^2 = 1.

Oh wow thank you! That's a much easier method!

However does m always equal 1? and k always equal 2a?

[Splash-Tackle-Flail]

Oh wow thank you! That's a much easier method!

However does m always equal 1? and k always equal 2a?

Don't quote me but from my year 11 knowledge (so I may be wrong!) 'm' is referring to the coordinate of the foci as bright sky said "let F_1 (-m,0) and F_2 (m,0)", which in your question -m refers to the -1 in |z-1| and m refers to the 1 in |z+1|. Thus m doesn't always equal one, and it just depends on the question.

Iirc k always equals 2a, unless its referring to a north south opening hyperbola in which case it will equal 2b.

Don't put this in a reference book or anything- best to get the opinion of someone who has already done spesh 3 4, this is just my poor attempt of helping in case no one else does (which is highly unlikely on AN )

[pi]

So would we be allowed to solve the question with that method (as in is loci in the spesh syllabus?)

Yeah I don't see why not, as long as you explain it logically it should be fine. Be wary if the question asks you to evaluate the equation "algebraically" though.

[pinklemonade]

Stuck on a question :/

If , express in polar form: