Co-Ordinate Geometry of the Circle
1. 2011 Paper 2 The line x + 3y = 20 intersects the circle x2 + y 2 − 6x − 8y = 0 at
the points P and Q. Find the equation of the circle that has [P Q] as diameter.
2. 2014 Paper 2
(a) The diagram shows a circular clock face, with the hands not shown. The square
part of the clock face is glass so that the mechanism is visible. Two circular
cogs, h and k, which touch externally are shown. The point C is the centre of
the clock face. The point D is the centre of the larger cog, h, and the point E
is the centre of the smaller cog, k.
i. In suitable co-ordinates, the equation of the circle h is x2 +y 2 +4x+6y−19 =
0 Find the radius of h, and the co-ordinates of its centre, D.
ii. The point E has co-ordinates (3, 2). Find the radius of the circle k.
iii. Show that the distance from C(−2, 2) to the line DE is half the length
of[DE].
iv. The translation which maps the midpoint of [DE] to the point C maps the
circle k to the circle j. Find the equation of the circle j.
v. The glass square is of side length l. Find the smallest whole number l such
that the two cogs, h and k, are fully visible through the glass.
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3. 2013 Paper 2 The circles c1 and c2 touch externally as shown.
Circle
c1
c2
Centre Radius
(−3, −2) 2
Equation
x2 + y 2 − 2x − 2y − 7 = 0
(a)
(b)
i. Find the co-ordinates of the point of contact of c1 and c2 .
ii. Hence,or otherwise, find the equation of the tangent,t, common to c1 and c2
4. 2012 Paper 2 The equation of two circles are:
c1 : x2 + y 2 − 6x − 10y + 29 = 0
c2 : x2 + y 2 − 2x − 2y − 43 = 0
(a) Write down the centre and radius-length of each circle.
(b) Prove that the circles are touching.
(c) Verify that (4, 7) is a point that they have in common.
(d) Find the equation of the common tangent.
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5. 2012 Paper 2 The circle shown in the diagram has, as tangents, the x − axis, the
y − axis, the line x + y = 2 and the line x + y = 2k, where k>1. Find the value of
k.
6. 2014 Sample Paper 2 The centre of a circle lies on the line x + 2y − 6 = 0. The
x − axis and the y − axis are tangents to the circle. There are two circles that satisfy
these conditions. Find their equations.
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