1. No category

CLASS X
CIRCLES
1. Two tangents PA and PB are drawn to the tangents with centre
C
O, such that ∠ APB = 1200. Prove that OP = 2AP.
2. In the given fig, AB is the diameter of a
circle with centre O and QC is a tangent
A
O
B
to the circle at C. If ∠ CAB = 300, find
(i) ∠ CQA, (ii) ∠ CBA.
B
3. In the given fig, PT is tangent to the circle
at T. If PA = 4 cm and AB = 5 cm, find PT.
Q
A
P
4. In the given fig, a circle touches the side
BC of ∆ ABC at P and touches AB and AC
produced at Q and R respectively. If AQ=
5 cm, find the perimeter of ∆ ABC.
T
A
P
B
C
R
Q
X
5. In fig, XP and XQ are two tangents to a circle
with centre O from a point X outside the circle.
ARB is tangent to the circle at R. Prove that
XA + AR = XB + BR
A
P
R
B
Q
.O
6. If ∆ ABC is isosceles with AB = AC and a circle with centre O is
the incircle of ∆ ABC touching BC at L. Prove that the point L
bisects BC.
7. ABC is a right angled triangle, right angled at A. A circle is
inscribed in it. The length of the two sides containing the right
angle are 12 cm and 5 cm. Find the radius of the incircle.
8. A circle is inscribed in a ∆ ABC having sides
8 cm, 10 cm and 12 cm as shown in fig. Find
E
F
AD, BE and CF. (Imp)
A
9. AB is diameter & AC is a chord of a circle such that ∠ BAC
is 300. If then tangent at C intersects AB produced in D.
Prove that BC = BD.
D
12 cm
8 cm
B
A’
Q
A
10. QR is a tangent at Q. PR ‖ AQ, where AQ
is a chord through A and P is the centre.
If AB is the diameter of the circle, prove
that BR is tangent to the circle.
R
P
B
B’
11. In a right triangle ABC, a circle with a side AB as diameter is
drawn to intersect the hypotenuse AC in P. Prove that the
tangent to the circle at P bisects the side BC.
B
12. In the given fig, AB is a diameter of the circle.
the length of AB = 5 cm. If O is the centre of the
C
O
circle and the length of tangent segment
AT = 12 cm. Determine CT.
A
13. AB is a diameter of a circle APB. AH and BK
are perpendicular from A and B respectively to the tangent at P.
Prove that AH + BK = AB.
14. In the given fig, the diameter of two wheels
B
A
have measures 2 cm and 4 cm. Determine
. O’
the lengths of the belts AD and BC that
P
.O
pass around the wheels if it is given that
C
D
belts cross each other at right angles.
15. AB is a chord of length 24 cm of a circle of radius 13 cm. The
tangents at A and B intersect at a point C. Find the length AC.
16. In the given fig, line AB is tangent to
A
B
both the circles touching at A and B.
OA = 29 cm, BP =18 cm, OP = 61 cm
. O’
O
then find AB.
m
B
l
D
17. In the given point A is a common point
C
of contact of two externally touching
.
A
. O’
circles and line l is a common tangent
O
to both circles touching at points B and
C. Line m is another common tangent at A and it
intersects BC at D. Prove that (a) ∠ BAC = 900 (b) Point D is
the midpoint of seg BC.
T
18. Two circles, one with centre M and
radius 4 cm and the other with centre
N and radius 6 cm, touch a line in points
P and Q respectively. M and N lie on
opposite sides of the line PQ. If MN =
20 cm, find PQ.
N
P
O
Q
19. If ∆ ABC is an equilateral triangle inscribed in a circle then
prove that the tangents at A, B and C form an equilateral
triangle.
20. Two circles, one with radius 10 cm and other with radius 8 cm
have the same centre P. Point C is on the larger circle.
Determine the length of a tangent segment from C to the smaller
circle.
21. A circle of radius 2 cm touches a circle of radius 10 cm
internally. Determine the length of a tangent segment drawn
through the centre of the larger circle to the smaller circle.
22. Two circles centred A and B touch internally at point C. The
tangents drawn through D to the circles touch at points M and
A
N, show that DN = DM.
23. In the given fig, the circle is the incircle of
isosceles ∆ ABC, where seg AB seg AC.
.
Prove that M bisects BC.
B
C
24. If four tangents of a circle determine a rectangle then show that
it must be a square.
M
25. In the adjoining fig, tangents at points
M and N intersect at P. ∠ MPN = 600,
.O
NM = 10 cm. Find (a) Radius of the circle
P
(b) Area of OMPN.
N
B
26. Lines l and m are tangents to
the circle with centres P and Q
as shown in fig. Show that
AB = CD.
A
.O
D
l
.O
C
m
27. Two circles with centres A, B are touching externally and a
circle with centre C touches both externally. If AB = 3 cm, BC =
3 cm and CA = 4 cm, find the radii of each circle.
28. In fig, points P and Q are the centres of the circle.
D
Radius QN = 3 cm, PQ = 9 cm. M is the point
C
of contact of the circles. Line ND is tangent
to the larger circle. Point C lies on the
.P
.Q
N
smaller circle. Determine NC, ND and CD.
M
29. Three congruent circles with centres A, B and C and with radius
5 cm each, touch each other in points D, E and F. Find the
perimeter of ∆ ABC, length of side DE of ∆ DEF.
30. Two circles which are not congruent touch externally. The sum
of their area is 130 cm2 and the distance between their centres
is 14 cm. Find radii of circles.
31. O is the centre and AB is diameter. Rays DB and DC are
tangents to the circles at B and C. Show that OD ‖ AC.
32. PQ is diameter. Line AQ, line AC and line CP are tangents to the
circle at point Q, B and P respectively. AQ = x, CP = y. Prove
that the radius of the circle is √
.
33. ABCD is a rectangle. Taking AD as a diameter, a semicircle AXD
is drawn which intersects the diagonal BD at X. If AB = 12 cm,
AD = 9 cm, find BD and BX.
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