WEBVTT

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Hi I'm Fiona and this is Tamura 2021
paper 2 and question 15.

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We are told that a circle has equation X
^2 plus AX plus y ^2 plus B y + C = 0

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where AB and C are non 0 real constants.

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And we're asked which one of the
following which is options A to F is a

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necessary and sufficient condition for
the circle to be tangent to the Y axis.

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Now,
when we think about whether something is

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necessary and sufficient,
we are looking for a condition that

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guarantees the result both ways.

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So I'm going to first start about start
to think about what does it mean for a

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circle to be tangent to the Y axis.

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When I think about a circle intersecting
the Y axis,

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there's only three possible ways that
this can happen.

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So here's my Y axis here,
and I could have a circle that doesn't

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intersect the Y axis at all.

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Here's another Y axis.

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I could have a circle that intersects the
Y axis at 2 distinct points.

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Or I could have a circle that intersects
the Y axis exactly once,

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and that could be from this side where
the the X coordinates on the circle are

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positive,
or from the negative X direction as well.

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So in order for there to be 1 distinct
point,

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which is what it would mean for the Y
axis to be a tangent to this circle,

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then I'm imagining A coordinate point
whose X value is 0 and whose Y value we

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don't know to lie on that circle.

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So I'm going to substitute this
coordinate point into this equation here

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and see where that gets me.

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So when X is 0 and I will end up with 0
^2 + a * 0 + y ^2 plus B y + C = 0.

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So that gives me a quadratic in YY
squared plus B y + C = 0.

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And this quadratic relates to these three
scenarios.

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And when we consider the discriminant of
the quadratic and so the discriminant of

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this quadratic would be b ^2 - 4 * a,
which here is 1 * C Now if the

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discriminant is less than 0,
we would have this scenario here.

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If the discriminant is greater than 0,
we would have this scenario here.

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And if the discriminant is equal to 0,
then we have what we want which is our

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circle touching or intersecting with the
Y axis at exactly 1 distinct point,

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thus making the Y axis a tangent to the
circle.

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So now I just need to rearrange this and
I could add 4C to both sides,

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which gives me b ^2 = 4 C and that is in
fact one of the conditions that I'm given.

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This condition is necessary and
sufficient in order for the circle to be

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tangent to the Y axis because this is the
only scenario in which we get that result.

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So the answer to this question is option
B,

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which is b ^2 = 4 C Let's take a moment
to reflect on this question.

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It's possible to approach this question
by considering the length of the radius

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of the circle and the X coordinate of the
centre of the circle.

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But this can introduce an element of
complexity in terms of the algebra that's

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involved in that,
as well as considering the fact that you

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would have a positive length for the
radius and you would need to be

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considering the negative X coordinate if
the circle was on the other side of the X

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axis here.

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And so this relationship between the
discriminant of this quadratic and the

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different scenarios that occur as a
result of the value of the discriminant

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can can give you a way to approach this
question that is efficient.

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And so you may be in other scenarios in
which you're thinking about the

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intersection of two mathematical objects
and you have a quadratic involved and

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thinking about the discriminant in that
scenario is very useful tool to have in

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your mathematical toolkit.