7.
Circles and tangents
When
we are able to find the algebraic equation of circles, it enables us to
solve important problems about the intersections of circles and other
curves using both our geometric knowledge about circles (e.g.
that the tangent to a circle is perpendicular to the radius) and our algebraic
knowledge of simultaneous equations (we can find the intersections by
solving the equations simultaneously).
Let’s
work through a few typical examples.
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Given
the circle x2
+ y2
= 25 and the point A(3, -4) on the circle. Find the equation
of the tangent to the circle at A by:
a.
using geometric knowledge about tangents.
b. using
algebraic knowledge about simultaneous equations.
Click
here to view the solution:  |
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The
line y
= x
+ c cuts the circle with centre at M(2, 3) and radius 4.
For what values of c:
a. is the line a secant to the circle?
b. is the line a tangent to the circle?
Click here to view a solution:
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The
line with equation 2x
– 3y
= 10 touches the circle with centre M(-2, 4) at the point
A. Find the equation of the circle and the coordinates of
A.
Click
here to view the solution: |
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Given the
circle x2 +
y2 = 16
and a point T(8, 2). Find the equation of the tangent from T
to the circle, and find the point of tangency in two
different ways:
- By
using knowledge about the geometric construction of the
tangent.
Descartes showed that “if it can be constructed,
it can be calculated”. Open the Tangent applet to see how
the tangent can be constructed, then use the idea to
solve the problem by calculation.
- By
using algebraic knowledge about simultaneous
equations.
Click on
the button to view the solution:
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