Class X · Mathematics Ch.10 · CBSE Pattern · 30 Questions
🟢 Section A — Multiple Choice 10 × 1 = 10 marks
Q1 · MCQ
The angle between a tangent to a circle and the radius at the point of contact is:
Q2 · MCQ
The number of tangents that can be drawn from a point inside a circle is:
Q3 · MCQ
From a point 10 cm from the centre of a circle of radius 6 cm, the tangent length is:
Q4 · MCQ
If PA and PB are tangents from an external point P with PA = 5 cm, then PB is:
Q5 · MCQ
A line intersecting a circle at two points is called a:
Q6 · MCQ
The number of tangents from a point on the circle is:
Q7 · MCQ
For a quadrilateral ABCD circumscribing a circle, which is true?
Q8 · MCQ
The tangents at the two endpoints of a diameter of a circle are:
Q9 · MCQ
If two tangents from P make an angle of 60°, the angle ∠AOB at the centre is:
Q10 · MCQ
A parallelogram circumscribing a circle is always a:
🟣 Section B — Assertion & Reason 3 × 1 = 3 marks
(a) Both A & R true, R explains A · (b) Both true, R doesn't explain A · (c) A true, R false · (d) A false, R true
Q11 · A–R
Assertion: The tangent at any point of a circle is perpendicular to the radius at the point of contact. Reason: The perpendicular is the shortest distance from a point to a line.
(a) Both true and R correctly explains A. The radius is the shortest (perpendicular) distance from the centre to the tangent.
Q12 · A–R
Assertion: Tangents drawn from an external point to a circle are equal in length. Reason: A tangent touches the circle at two points.
(c) Assertion is true, but the Reason is false — a tangent touches the circle at exactly ONE point, not two.
Q13 · A–R
Assertion: Two tangents can be drawn to a circle from an external point. Reason: A point outside a circle lies farther from the centre than the radius.
(a) Both true and R explains A — because the point is outside (distance > radius), exactly two equal tangents exist.
🟢 Section C — Short Answer 12 × 2 = 24 marks
Q14 · Short
A tangent is drawn from a point 25 cm from the centre of a circle of radius 7 cm. Find its length.
Prove that the tangents drawn from an external point to a circle are equal.
In △OAP and △OBP: OA = OB (radii), OP common, ∠OAP = ∠OBP = 90°. By RHS, △OAP ≅ △OBP, so PA = PB (CPCT).
Q17 · Short
A quadrilateral ABCD circumscribes a circle. If AB = 6, BC = 7, CD = 4, find DA.
AB + CD = BC + DA → 6 + 4 = 7 + DA → DA = 3 units.
Q18 · Short
Two tangents from P touch a circle at A and B. If ∠APB = 70°, find ∠OAB (O is the centre).
∠AOB = 180° − 70° = 110°. △OAB is isosceles (OA=OB), so ∠OAB = (180−110)/2 = 35°.
Q19 · Short
Define: (a) tangent to a circle, (b) point of contact.
(a) A tangent is a line touching the circle at exactly one point. (b) The point of contact is that single common point of the tangent and the circle.
Q20 · Short
The length of a tangent from a point 17 cm from the centre is 15 cm. Find the radius.
r = √(d²−ℓ²) = √(17²−15²) = √(289−225) = √64 = 8 cm.
Q21 · Short
A tangent to a circle of radius 6 cm from a point P has length 8 cm. Find the distance OP.
OP = √(r²+ℓ²) = √(6²+8²) = √(36+64) = √100 = 10 cm.
Q22 · Short
If two tangents from an external point are inclined at 80° to each other, find the angle each tangent makes with the line joining P to the centre.
OP bisects ∠APB, so each half = 80°/2 = 40°.
Q23 · Short
Prove that a parallelogram circumscribing a circle is a rhombus.
AB+CD = BC+DA. In a parallelogram AB=CD and BC=DA, so 2AB = 2BC → AB = BC. All sides equal ⇒ rhombus.
Q24 · Short
How many tangents can be drawn to a circle that are parallel to a given secant? Explain.
Two — one on each side of the circle, both parallel to the given line and touching the circle at diametrically opposite points.
Q25 · Short
Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the larger circle that touches the smaller.
The smaller circle bisects the chord. Half-chord = √(5²−3²) = √16 = 4. Full chord = 8 cm.
🔴 Section D — Long Answer 5 × 2 = 10 marks
Q26 · Long
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Let XY be a tangent at P, O the centre. Take any other point Q on XY. Q lies outside the circle, so OQ > OP. Thus OP is the shortest distance from O to XY, and the shortest distance to a line is the perpendicular. Hence OP ⊥ XY.
Q27 · Long
A circle touches all four sides of quadrilateral ABCD. Prove AB + CD = BC + DA.
Let tangent lengths from A,B,C,D be p,q,r,s. Then AB = p+q, BC = q+r, CD = r+s, DA = s+p. AB+CD = p+q+r+s and BC+DA = q+r+s+p. Both equal ⇒ AB+CD = BC+DA.
Q28 · Long
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.
A circle of radius 4 cm is inscribed in a triangle. A tangent from vertex A has length 8 cm. If the other two tangent lengths are 6 cm and 10 cm, find the perimeter.
Each side = sum of two tangent segments. Sides = (8+6), (6+10), (10+8) = 14, 16, 18. Perimeter = 48 cm.
Q30 · Long
PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q meet at T. Find TP.
Perpendicular from O bisects PQ at M: PM = 4, OM = √(5²−4²) = 3. Let TP = TQ = x, TM = √(x²−16). Using similar triangles OP/PT = OM/... → solve: TP = 20/3 cm ≈ 6.67 cm.