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NARRATOR: Exercise 12.

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Show that the line y = –2x + 5

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is tangent to the circle
x squared + y squared = 5.

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If the line is tangent to the circle,

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then there is only
one point of intersection

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between the line and the circle.

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The point or points of intersection
can be found

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by substituting the linear equation
into the equation for the circle,

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giving the following.

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Expanding –2x + 5 all squared
and then collecting like terms

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gives a quadratic expression.

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Dividing through by 5,
then factorising and then solving

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shows that x = 2 is the only solution
to this equation.

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So since there is only
one point of intersection

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between the line and the circle,
at the point where x = 2,

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the line must be tangent
to the circle.

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At this point of the process,
we could also have shown that

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the discriminant of this quadratic
is equal to 0,

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and hence there is only one solution
to this equation

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that gives the points
of intersection,

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and therefore only
one point of intersection.

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But in this case, it was very simple
to solve this quadratic

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once we reach this point.