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December 08, 2019, 11:25:38 am

Author Topic: SAC EMERGENCY HELP!  (Read 387 times)  Share 

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withez

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SAC EMERGENCY HELP!
« on: May 25, 2019, 09:45:58 am »
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Hi everyone
Sorry if this is a bit rambly.
I have done Year 11 methods in Year 10, and I have the same teacher this year for Year 11 Specialist.
Yesterday we had a problem-solving assessment (which I didn't have time to prepare for because I have Year 11 exams plus heavily weighted year 12 assessment tasks in two subject). My spesh teacher has been giving me  Year 12 spesh work because we "should have covered what we are doing in 10 advanced", so I didn't have time to look back and see what the rest of the class was doing.
The problem solving task had bearings, ellipses, parametric equations and modulus (both of which he said wouldn't be assessed!) which we haven't looked at for months/years.
I also had a minor migraine in the middle of it, so basically I finished one question and started the calculations for a few others. We have one more lesson on Monday.
I have a Unit 3 Bio SAC this Thursday and shortly after a Unit 3 Methods Application task which is going to be basically all of my Unit 3 SAC mark, so I don't have time to prepare.
I would be so grateful if someone could help me with what I remember of the task:
How do you prove that x^2 +y^2+a^2+2axy= 1 touches the x axis (or y axis- can't remember which axis the question referred to)?
How do you find tangents to circles?
Could someone explain how to solve modulus equations if they are literal (I don't remember much more about the question than that unfortunately)
How do you find the range of an ellipse?
Is there some "secret" to solving complex bearings problems?
How do you do parametric equations?
Thank-you so much

AlphaZero

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Re: SAC EMERGENCY HELP!
« Reply #1 on: May 25, 2019, 03:09:51 pm »
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Hey there.

May is always a busy month for most VCE students, so hang in there - it's only temporary ;)

Some advice:  Do remember that for this year, you have two subjects that will contribute to your ATAR, and so those should be your priority. Note that I'm NOT saying you should neglect your year 11 studies. I'm just trying to remind you of what is important, so do make sure you are feeling prepared and confident for Methods and Bio first.

How do you find the range of an ellipse?

Consider the following relation: \[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,\quad a,b>0.\] This is an ellipse that
> is centered at \((h,\,k)\),
> has a horizontal semi-axis length of \(a\),  and
> has a vertical semi-axis length of \(b\).

From the following image, it's quite easy to find the domain and range.


How do you prove that x^2 +y^2+a^2+2axy= 1 touches the x axis (or y axis- can't remember which axis the question referred to)?

Unfortunately, I'm not sure you remembered the relation correctly, since for \[C:\ \ x^2+y^2+a^2+2axy=1,\] the graph either crosses the coordinate axes, or doesn't touch the coordinate axes at all (depending on the value of \(a\)). That is, for all \(a\in\mathbb{R}\), the graph never touches either axis without crossing.

There are a few ways you can go about proving that a relation only touches either coordinate axis. For example, for conic sections such as circles and ellipses, you could find the 'domain' and range. Take the graph of  \((x-2)^2+y^2=4\), which is a circle of radius 2 centered at \((2,\,0)\). So, it has domain \([0,\,4]\) and range \([-2,\, 2]\),  and therefore touches (but does not cross) the \(y\)-axis.

How do you find tangents to circles?

There are a couple ways to go about this. Let \(C\) denote the centre of the circle and let \(P\) be the point on the circle where the tangent touches. The simplest way to find the equation of the tangent is to realise that the tangent will be perpendicular to the line segement \(CP\).

For example, let's find the equation of the tangent to the graph of  \((x-2)^2+y^2=4\)  at the point  \(P(1,\,\sqrt{3})\).  The gradient of the line joining \(P\) and the centre of the circle \(C(2,\,0)\) is \[m_{CP}=\frac{\sqrt{3}-0}{1-2}=-\sqrt3,\] and so the gradient of the tangent to the graph at \(P\) is \(\dfrac{1}{\sqrt{3}}\). Hence, the equation of the tangent is \[y-\sqrt{3}=\frac{1}{\sqrt{3}}(x-1)\\
\implies y=\frac{1}{\sqrt{3}}x+\frac{2\sqrt3}{3}.\]
Could someone explain how to solve modulus equations if they are literal (I don't remember much more about the question than that unfortunately)

Recall that \[|x|=\begin{cases}x, & x\geq 0\\
-x, & x<0\end{cases}.\] You can essentially break up the problem into cases. Suppose we want to solve \[|ax-b|=c,\quad a>0,\ \ b\in\mathbb{R},\ \ c\geq 0.\] Then, using our definition, we have \[ax-b=c\ \text{ if }x\geq\frac{b}{a}\quad\text{and}\quad b-ax=c\ \text{ if }x<\frac{b}{a}\\
\implies x=\frac{b+c}{a}\ \text{ or }\ x=\frac{b-c}{a}.\] Alternatively, you can use \(|x|=\sqrt{x^2}\) if it makes it easier (note that sometimes, it won't).

Is there some "secret" to solving complex bearings problems?

Not really. You get more efficient at them with practice. It's not like there's a trick so that you can see solutions immediately unfortunately.

How do you do parametric equations?

To be honest, I'm not sure what you mean by "do" parametric equations. There are lots of things you can do with them. What specifically are you looking for?

Hope this has been helpful :)
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withez

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Re: SAC EMERGENCY HELP!
« Reply #2 on: May 26, 2019, 08:24:48 am »
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Thankyou so much AlphaZero
It makes so much more sense to me now.
Don't worry about parametric equations, I have revisited them and I remember everything about them now.
You were right- i didn't recall the circle equation correctly. I think I misread it in the task, hence why I was having so much trouble with it.
Again, thanks for all of the help, you are a lifesaver.