WEBVTT

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Hi, I'm Fiona.

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We're going to have a look at TAMUA 2021
paper one and question 18.

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We are given a curve with equation X = y
^2 -,

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6 Y plus 11 and we're told that this
curve is rotated 90° clockwise about the

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point P to give the curve.

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CP has X coordinate -2 and Y coordinate 3
and we are asked what is the equation of

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this curve?

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See.

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Now the first thing I notice is this
equation that I'm given.

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I'm not as familiar with equations of X
in terms of Y as I am with equations of Y

62855bbd-b70a-4884-9ab9-72e8c2e91f24-1
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in terms of X.

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So I'm going to 1st swap the X and YS and
see where I can go from there.

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This is now something I'm much more
familiar with.

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This is a quadratic in XI.

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Could think about factorising it to find
the roots.

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I could think about finding its Y
intercept.

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What I'm going to do is get it in its
completed square form and take it from

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there.

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So for this quadratic in its completed
square form, it would be X -,

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3 all squared.

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And then if I were to multiply out this
bracket, the constant term would be a +9.

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So I'm going to take 9 away to keep it
balanced and then bring down this plus 11.

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So that is going to give me a completed
for completed square form of X -,

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3 all squared +2.

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So that tells me that this is a quadratic
with a minimum at the .32.

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Let's draw that.

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Here's a rough sketch of the quadratic.

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Now I know that the equation that or the
curve that I actually want to graph has

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this equation.

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So in order to get that curve,
what I'm going to do is reverse my

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swapping of X's and Y's.

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Now graphically,
this means that I'm going to be

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performing a reflection in the line y = X.

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But really I'm just going to think about
this point.

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If I swap the X's and Y's then this point
on my curve will now be have an X value

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of two and AY value of three.

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Let's draw that.

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Here's my new point which is in fact a
vertex of a parabola.

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With this orientation you can picture the
reflection in the line y = X.

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And that's all the information we need to
know to be able to get a rough idea of

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this curve that we wanted to graph.

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Now,
this is the curve that is going to be

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rotated 90° clockwise about the point PP
has coordinates -2 three.

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So P is there.

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Let me just label it with a big P.

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Now if I were to rotate this curve 90°
clockwise about,

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Pi can see that it would rotate around
like this.

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And this point here,
this vertex here would remain a distance

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4 away from P but be vertically below P
So that means that it would have AY

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ordinate of -1 and it would be the point
-2 -, 1.

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And as I can also imagine,
this curve rotating around clockwise 90°

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and it would look something like this.

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Now this is a quadratic with a negative
coefficient of X ^2 because it's got that

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N shape and with a maximum at -2 -, 1.

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So I can form its equation by starting
with the completed square form.

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So that would be Y equals minus to get
that negative coefficient of X ^2 for the

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N shape.

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And then I would have X + 2 in the
brackets all squared,

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and then I would have a -1 here.

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And now I'm going to multiply that out to
get the equation of this curve C that

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I've been asked to find.

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And here's the equation of the curve I've
been asked to find.

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Y = -, X ^2 -, 4 X -5.

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Now,
looking at the options of the given to

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choose from to answer this question,
I can see that B has a constant term of

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-5.

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Do all the other terms match?

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Yes, they do.

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So B is the answer to this question.

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Let's take a moment to reflect on this
question.

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I was given an equation of X in terms of
Y and I wasn't as familiar with that as

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the equation that I got when I swapped
the X's and Y's.

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And this little trick of swapping X's and
Y's in an equation and what that does

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graphically is handy to have in mind.

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So when I got my quadratic graph here,
which was very familiar to me,

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in order to kind of swap the graph back
to get the curve with this equation,

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I was having in my mind that this is the
same as performing a reflection in the

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line y = X.

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And that is handy to keep in mind as a
technique in your mathematical toolkit.

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It certainly came in handy in this
question,

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this curve here with this equation of X
in terms of Y.

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It's still a parabola,
just like our quadratic,

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but it has a different orientation.

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And you learn more about these kinds of
curves under the topic of conics.

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So if you're interested,
you can look up the topic of conics to

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find out more.