WEBVTT

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Hi, I'm Fiona.

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Let's have a look at TMUA 2021 paper one
and question one.

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We're told 2 circles have the same radius.

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The centre of 1 circle is -2 one and the
centre of the other circle is 3 -, 2.

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We're told that the circles intersect at
2 distinct points,

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and we're asked what is the equation of
the straight line through the two points

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at which the circles intersect.

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So here we've got 2 circles.

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We know they're centres,
we know that they intersect,

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and they intersect at 2 distinct point,
which means they're not just touching,

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they're actually overlapping.

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The key here is that the two circles have
the same radius.

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So let me draw what's going on.

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2 circles that have the same radius and
in order to find the straight line

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through the two points at which they
intersect, which would be this line here.

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This line here is the perpendicular
bisector of the line segment between the

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two circle centres.

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Now in order to find the line I need,
which is this one here,

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I need 2 points that lie on that line or
I need one point and a gradient.

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Now an obvious point that lies on this
line here is the midpoint of this line

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segment between the two circle centres.

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So to get the midpoint,
I'm going to get the average of the X

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ordinates which is 3 -, 2 / 2,
so that's a half and the average of the Y

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ordinates which is -2 + 1 / 2.

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So that's minus 1/2.

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And then I need to get a gradient.

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So if I get the gradient of this line
segment,

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then I'm going to take the negative
reciprocal in order to find the gradient

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of this perpendicular bisector.

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So the gradient of this line segment.

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Let me do it over here.

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I get the difference in Y values, so -2 -,
1,

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and I get the difference in X values on
the bottom.

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So 3 -, -, 2, that's going to be 3 + 2.

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So that gradient there is -3 / 5.

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And that means that the gradient that I'm
looking for,

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for the line that I need is going to be
the negative reciprocal of that.

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So that's going to be a gradient of 5 / 3.

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And now I'm going to use my formula of Y
minus Y 1 = m on X -, X one.

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So Y 1 is going to be this value Here y -,
y one is y + 1/2 M is 5 / 3 on X -,

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X one which is 1/2.

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So that's minus 1/2.

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Tidying up this line here, I get y = 1/2,
not y = 1/2 Y plus 1/2 = 5 / 3 X -5 / 6,

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and I'm going to multiply all across by 6
to get rid of those denominators.

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Let's just bring it over here a bit.

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So I will get six y + 3 = 10 X -5.

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I'm looking over at the options to see
which form my answer needs to be in.

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I need my X term first,
followed by my Y term equals and then

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constant term.

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So that gives me -10 X plus six y = -,
5 - 3,

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which is going to be -8 and I'm going to
divide all across by -2 which will give

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me five X - 3 Y equals a +4.

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And having a look at the options,
I see that option F gives me that answer

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5X minus three y = 4.

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And so my answer is option F for this
question.