Chapter 10: Circles
The Complete 59-Part Master PYQ Archive
π 59 Unique Questions
π― 5-Pillar Solutions
✍️ 100% Board Mapped
π Phase 1: CBSE Sample Question Papers (2022 - 2027)
Part 1 | Proof
#01
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ = 2\angle OPQ$.
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Part 2 | Proof
#02
XY and X'Y' are parallel tangents. Another tangent AB with point of contact C intersects XY at A and X'Y' at B. Prove that $\angle AOB = 90^\circ$.
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Part 4 | 5 Marks
#04
Prove tangent lengths are equal. Using this, find length of BC for a circle inscribed in $\Delta ABC$ with given side lengths $AB=10$, $AQ=7$, $CQ=5$.
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Part 5 | 2 Marks
#05
PA and PB are tangents from P. A tangent at E intersects PA and PB at C and D. If $PA = 10\text{ cm}$, find the perimeter of $\Delta PCD$.
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Part 6 | 2 Marks
#06
In the given figure, O is the centre. Find $\angle AQB$, given that PA and PB are tangents and $\angle APB=75^\circ$.
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Part 7 | 2 Marks
#07
$\Delta ABC$ circumscribes a circle of radius $4\text{ cm}$. $BD=10$, $DC=8$. If $\text{Area}(\Delta ABC) = 90\text{ cm}^2$, find AB and AC.
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Part 8 | Case Study
#08
Meditation garden. Tangents PA and PB drawn from P. Radius $r=9\text{ m}$, distance $OP=18\text{ m}$. Find $\angle OAP$, length PA, and $\angle AOB$.
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Part 9 | 1 Mark MCQ
#09
If two tangents inclined at an angle of $60^\circ$ are drawn to a circle of radius $3\text{ cm}$, find the length of each tangent.
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Part 10 | Proof
#10
Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that $\angle APB = 2\angle OAB$.
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Part 11 | 1 Mark MCQ
#11
A quadrilateral ABCD circumscribes a circle. If $BC=7\text{ cm}$, $CD=4\text{ cm}$ and $AD=3\text{ cm}$, find the length of AB.
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Part 12 | 1 Mark MCQ
#12
Find the length of a tangent drawn to a circle of radius $9\text{ cm}$ from a point at a distance of $41\text{ cm}$ from the centre.
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Part 13 | 1 Mark MCQ
#13
Chord PQ makes an angle of $50^{\circ}$ with the tangent PR at the point of contact P. Find the angle subtended by the chord at the centre.
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Part 14 | 1 Mark MCQ
#14
A tangent is drawn at point P on a circle centred at O. If $\angle TPQ = 110^{\circ}$, find the value of $\angle POQ$.
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Part 15 | 1 Mark MCQ
#15
O is the intersection of chords AB and CD. If $\angle BOC=80^{\circ}$ and $OA=OD$, determine the relationship between $\Delta ODA$ and $\Delta OBC$.
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Part 16 | 1 Mark MCQ
#16
Quadrilateral PQRS circumscribes a circle. If $PQ = 12\text{ cm}$, $QR = 15\text{ cm}$ and $RS = 14\text{ cm}$, find the length of SP.
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Part 17 | 1 Mark MCQ
#17
Identify the false statement regarding the theoretical properties of tangents, secants, and radii of a circle.
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Part 18 | 3 Marks
#18
Tangents PA, PB drawn. Chord AB makes $30^{\circ}$ with the radius. If length of chord is $6\text{ cm}$, find tangent PA and radius OA.
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Part 19 | 2 Marks
#19
A circle is inscribed in a right-angled triangle ABC, right angled at B. If $BC = 7\text{ cm}$ and $AB = 24\text{ cm}$, find the radius.
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π Phase 2: Official CBSE Board Exams (Foundation)
Part 20 | 1 Mark MCQ
#20
Circles O and N touch at P. Radius O is twice N. Tangent $OX=18\text{ cm}$. Find radius of N.
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Part 21 | 1 Mark MCQ
#21
Tangents P, T, S are drawn. If tangent base $QR=12\text{ cm}$ and $r=7\text{ cm}$, find the perimeter of polygon PQRSO.
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Part 22 | 1 Mark MCQ
#22
Tangents PA and PB are perpendicular. If chord $AB=5\sqrt{2}\text{ cm}$, find the length of tangent PA.
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Part 23 | 1 Mark MCQ
#23
AT is a tangent, AB is diameter. If interior $\angle ABC=50^{\circ}$, find the alternate segment angle $\angle CAT$.
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Part 24 | 2 Marks
#24
Prove that if a quadrilateral is circumscribed about a circle, the angles subtended by opposite sides at the centre are supplementary.
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Part 25 | 2 Marks
#25
PQ is tangent, QOR is diameter. If external angle $\angle POR=120^{\circ}$, find the tangent angle $\angle OPQ$.
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Part 26 | Case Study
#26
$\Delta ABC$ circumscribes a circle ($r=4$). Tangent point divides AB into $6$ and $8$. Area is $84$. Find AC and BC.
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Part 27 | 2 Marks
#27
AB and CD are tangents intersecting the chord AC. Is $\angle BAC = \angle DCA$? Justify using geometry.
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Part 28 | 2 Marks
#28
Radii OA and OB are perpendicular. Tangents AT, BT intersect at T. Prove $AB=OT$ and that they bisect each other.
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Part 29 | Case Study
#29
Ramesh spins a ball ($r=2$). Slips at $60^\circ$ tangent. Circumscribed quad $AB=3$, $AD=BC=5$. Find diameter, tangent, and CD.
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Part 30 | 2 Marks
#30
AB is diameter, AC is chord ($\angle BAC=30^\circ$). Tangent at C intersects AB extended at D. Prove $BC=BD$.
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Part 31 | 5 Marks
#31
Tangents TP and TQ to circle M touch circle N at A and B. $MQ=13$, $NB=8$, $BQ=35$, $TP=80$. Find lengths TB and MN.
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Part 32 | 5 Marks
#32
PQ and PR are tangents ($r=5$). AB is tangent at C on OP. Distance $OP=13$. Find length of AB and PA.
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Part 33 | 3 Marks
#33
A circle touches QR at S and extended sides PQ and PR at M and N. Prove that $PM = \frac{1}{2}(PQ+QR+PR)$.
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Part 34 | 3 Marks
#34
$\Delta ABC$ is a right triangle where $\angle B=90^{\circ}$, $AB=4\text{ cm}$, and $BC=3\text{ cm}$. Find the exact radius of the inscribed circle.
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Part 35 | 2 Marks
#35
Two concentric circles are of radii $5\text{ cm}$ and $4\text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle.
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Part 36 | 2 Marks
#36
A circle is inscribed inside $\Delta LMN$ with tangency points A, B. If the reflex $\angle AOB = 240^{\circ}$, find the measure of $\angle ANB$.
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Part 37 | 3 Marks
#37
If a regular hexagon ABCDEF circumscribes a circle, then prove mathematically that $AB+CD+EF = BC+DE+FA$.
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π₯ Phase 3: The Final Bosses (2023 - 2026 Variations)
Part 38 | 3 Marks
#38
From point P, two tangents PA and PB are drawn. If $OP$ is equal to the diameter, prove $\Delta PAB$ is an equilateral triangle.
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Part 39 | 1 Mark MCQ
#39
In a circle, a chord MN and a tangent ML make an angle of $70^{\circ}$. Find the central angle $\angle MON$.
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Part 40 | 1 Mark MCQ
#40
The sum of radii of concentric circles is $16\text{ cm}$. The length of the tangent chord is $16\text{ cm}$. Find the difference of the radii.
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Part 41 | 1 Mark MCQ
#41
PA and PB are tangents such that $\angle P=90^{\circ}$. If chord $AB=3\sqrt{2}\text{ cm}$, find the exact diameter of the circle.
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Part 42 | 1 Mark MCQ
#42
TS is a tangent. In $\Delta OST$, $\angle O = 3x^{\circ}$ and $\angle T = x^{\circ}$. Find the algebraic value of $2x^{\circ}$.
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Part 43 | 1 Mark MCQ
#43
AB is a tangent chord. If length of $AB = \text{diameter of inner circle} = 2r$, find the diameter of the larger circle.
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Part 44 | 3 Marks
#44
Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.
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Part 45 | 3 Marks
#45
A circle ($r=10\text{ cm}$) is inscribed in a rectangle. Prove the rectangle is a square and calculate its perimeter.
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Part 46 | 3 Marks
#46
From external point P, tangents PA and PB are drawn. Prove that OP is the perpendicular bisector of chord AB.
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Part 47 | 1 Mark MCQ
#47
MN is a chord and tangent ML makes an angle of $70^{\circ}$ with MN. Find the central angle $\angle MON$.
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Part 48 | 1 Mark MCQ
#48
AC is the diameter of a larger circle. Tangent AB touches the inner circle at D. If inner $OD=r$, find the length of BC.
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Part 49 | 1 Mark MCQ
#49
Tangent PT is drawn such that $\angle TPO = 35^{\circ}$. Find the exterior obtuse angle $x$ formed at the center O.
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Part 50 | 1 Mark MCQ
#50
A parallelogram having one of its sides $5\text{ cm}$ circumscribes a circle. Find the perimeter of the parallelogram.
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Part 51 | 2 Marks
#51
If two tangents inclined at an angle of $60^{\circ}$ are drawn to a circle of radius $3\text{ cm}$, find the length of each tangent.
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Part 52 | 1 Mark MCQ
#52
AB and AC are tangents to a circle. If the base angle $\angle ABC = 42^{\circ}$, find the apex angle $\angle BAC$.
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Part 53 | 1 Mark MCQ
#53
Two circles touch externally. A common tangent is drawn. A vertical tangent AP of $4.2\text{ cm}$ is drawn. Find the common tangent length.
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Part 54 | 1 Mark MCQ
#54
RJ and RL are two tangents. If the tangent-chord angle $\angle RJL = 42^{\circ}$, find the central angle $\angle JOL$.
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Part 55 | 3 Marks
#55
AB is a diameter. AQ, BP, and PQ are tangents. Prove that the intersecting angle $\angle POQ = 90^{\circ}$.
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Part 56 | Case Study
#56
A backyard is a right $\Delta ABC$ ($\angle B=90^{\circ}$) with $AB=7\text{ m}$, $BC=15\text{ m}$. A circle is inscribed. Calculate its exact radius.
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Part 57 | 1 Mark MCQ
#57
PA and PB are tangents from P. Q is a point on the major arc. If $\angle APB = 55^{\circ}$, find the major arc angle $\angle AQB$.
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Part 58 | 1 Mark MCQ
#58
PQ is tangent at B. Central angle $\angle AOB = 95^{\circ}$. Find the tangent-chord angle $\angle ABQ$.
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Part 59 | 2 Marks
#59
PA is a tangent and PBC is a secant passing through centre O. If $\angle AOC=130^{\circ}$, find the exterior angle $\angle APB$.
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