PROBLEMS
03

CIRCLE
Page
1
( 1 )
Obtain
the
equation
of
the
circle
passing
through
the
points
( 5,
 8 ),
(  2,
9 )
and
( 2,
1 ).
[ Ans:
x
2
+
y
2
+
116x
+
48y

285
=
0 ]
( 2 )
Find
the
equation
of
the
circumscribed
circle
of
the
triangle
formed
by
three
lines
given
by
x
+
y
=
6,
2x
+
y
=
4
and
x
+
2y
=
5.
[ Ans:
x
2
+
y
2

17x

19y
+
50
=
0 ]
( 3 )
Find
centre
and
radius
of the
circle
whose
equation
is
4x
2
+
4y
2

12x
+
24y
+
29
=
0.
2
,
3
,
2
3
:
Ans

( 4 )
Find
the
equation
of
the
circle
touching
both
the
axes
and
passing
through
( 1,
2 ).
[ Ans:
x
2
+
y
2

2x

2y
+
1
=
0,
x
2
+
y
2

10x

10y
+
25
=
0 ]
( 5 )
The
cartesian
equation
of
the
circle
is
x
2
+
y
2
+
4x

2y

4
=
0.
Find
its
parametric
equations.
[ Ans:
x
=
 2
+
3 cos
θ
,
y
=
1
+
3 sin
θ
,
θ
∈
( 
π,
π
] ]
( 6 )
Show
that
the
linesegments,
joining
any
point
of
a
semicircle
to
the
end
points
of
the
diameter,
are
perpendicular
to
each
other.
( 7 )
Show
that
the
point
( 4,
 5 )
is
inside
the
circle
x
2
+
y
2

4x
+
6y

5
=
0.
Find
the
point
on
the
circle
which
is
at
the
shortest
distance
from
that
point.
[ Ans:
( 5,
 6 ) ]
( 8 )
Find
the
length
of
the
chord
of
the
circle
x
2
+
y
2

4x

2y

20
=
0
cut
off
by
the
line
x
+
y

10
=
0.
[ Ans:
2
]
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03

CIRCLE
Page
2
( 9 )
Find
the
midpoint
of
the
chord
of
the
circle
x
2
+
y
2
=
16
cut
off
by
the
line
2x
+
3y

13
=
0.
[ Ans:
( 2,
3 ) ]
( 10 )
Find
the
conditions
for
the
line
x cos
α
+
y sin
α
=
p
to
be
the
tangent
to
the
circle
x
2
+
y
2
=
r
2
.
[ Ans:
p
2
=
r
2
]
( 11 )
Find
the
equations
of
the
tangents
to
the
circle
x
2
+
y
2
=
17
from
the
point
( 5,
3 ).
[ Ans:
4x

y

17
=
0,
x
+
4y

17
=
0 ]
( 12 )
The
lengths
of
the
tangents
drawn
from
a
point
P
to
two
circles
with
centre
at
origin
are
inversely
proportional
to
the
corresponding
radii.
Show
that
all
such
points
P
lie
on
a
circle
with
centre
at
origin.
( 13 )
Find
the
measure
of
an
angle
between
two
tangents
to
the
circle
x
2
+
y
2
=
a
2
drawn
from
the
point
( h,
k ).
[ Ans:
2 tan

1
( a
/
2
2
2
a
k
h

+
) ]
( 14 )
Find
the
set
of
all
points
P
outside
a
circle
x
2
+
y
2
=
a
2
such
that
the
tangents
to
the
circle,
drawn
from
P,
are
perpendicular
to
each
other.
[ Ans:
x
2
+
y
2
=
2a
2
]
( 15 )
Find
the
equation
of
the
circle
which
passes
through
the
points
of
intersection
of
the
circles
x
2
+
y
2
=
13
and
x
2
+
y
2
+
x

y

14
=
0
and
whose
centre
lies
on
the
line
4x
+
y

6
=
0.
[ Ans:
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 Spring '06
 Jones

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