Class 9 Mathematics Chapter 10 Circles Previous Year Questions: Tangent, Length & Proof PYQs (2020–2025)

Reading Chapter 10 twice doesn't cement your understanding. Solving a real CBSE question on 'tangent from an external point' does. Here's why: First, PYQs reveal what examiners prioritize. Over the last 5 years, nearly 70% of Chapter 10 questions focus on tangent length calculation and proof—not theoretical definitions. Second, working under time pressure trains your brain to recognize patterns instantly. When you see 'two tangents from point P to circle centre O', you automatically visualize the right triangle and know to apply Pythagoras. Third, past papers expose your gaps. A student might think they understand the theorem, but struggle when asked to prove that 'the tangent to a circle is perpendicular to the radius at the point of contact.' Solving PYQs forces you to articulate that proof step-by-step. Fourth, mark schemes teach you what 'full marks' looks like. A 5-mark proof question isn't graded on your final answer alone—it's on construction clarity, logical flow, and notation. PYQs with detailed solutions show exactly how to write your answer for maximum marks. Finally, confidence compounds. Each PYQ you solve reduces exam anxiety. By the time you walk into the board exam, tangent problems feel familiar, not frightening.

These bite-sized questions test conceptual recall and quick geometry recognition. They're easy marks if you know the definitions—and easily lost if you don't. Study these five patterns closely: **Q1: How many tangents can be drawn to a circle from an external point?** Answer: Two. When a point lies outside a circle, exactly two tangent lines touch the circle. Both tangents are equal in length. This is a property, not a proof—memorize it. **Q2: What is the angle between a tangent and a radius at the point of tangency?** Answer: 90° (or a right angle). The radius drawn to the point where a tangent touches the circle is always perpendicular to that tangent. This is fundamental to all tangent-length calculations. **Q3: A line touches a circle at exactly one point. What is this line called?** Answer: Tangent (or tangent line). A tangent does not pass through the interior of the circle. **Q4: How many tangents can be drawn from a point on the circle itself?** Answer: One. When the point is on the circle, only one tangent line exists at that point. **Q5: If PT is a tangent to circle with centre O, and OP = 13 cm, PT = 5 cm, find the radius.** Answer: 12 cm. Using Pythagoras: OT² + PT² = OP², so OT² + 25 = 169, giving OT = 12 cm. These 1-mark questions often cluster in the 'objective' or 'multiple-choice' section. Each is testable as a standalone definition or instant calculation.

3-mark questions demand one key geometric property, a diagram, and either a proof outline or a multi-step calculation. Here are five patterns that appear consistently: **Q1: Prove that tangents drawn from an external point to a circle are equal in length.** Step-by-step answer: Let PA and PB be tangents from external point P to circle centre O. Join OA, OB, and OP. In triangle OAP and triangle OBP: OA = OB (radii), angle OAP = angle OBP = 90° (radius ⊥ tangent), OP = OP (common). By RHS (right angle–hypotenuse–side), △OAP ≅ △OBP. Therefore PA = PB. (Award marks for clear diagram, naming triangles, citing congruence rule, and final statement.) **Q2: Two tangents PA and PB are drawn to a circle from external point P. If angle APB = 60°, find angle AOB (O is centre).** Answer: In quadrilateral OAPB, angle OAP = 90°, angle OBP = 90°, angle APB = 60°. Sum of angles in quadrilateral = 360°. So angle AOB = 360° − 90° − 90° − 60° = 120°. **Q3: PT is a tangent to a circle with centre O and radius 5 cm. If OP = 13 cm, find the length PT. Also verify that angle OTP = 90°.** Answer: In right triangle OTP, OT² + PT² = OP². So 25 + PT² = 169, giving PT = 12 cm. Since OT = 5 (radius) and OP = 13, and 5² + 12² = 13², the Pythagorean relation holds, confirming angle OTP = 90°. **Q4: From an external point, two tangents of length 8 cm each are drawn to a circle of radius 6 cm. Find the distance of the external point from the centre.** Answer: Let P be external, PA = PB = 8 (tangents), OA = OB = 6 (radius). Since PA ⊥ OA, triangle OAP is right-angled. OP² = OA² + PA² = 36 + 64 = 100, so OP = 10 cm. **Q5: A tangent to a circle is perpendicular to the radius at the point of tangency. Justify why a line cannot be both a tangent and a secant to the same circle at the same point.** Answer: A tangent touches the circle at exactly one point and is perpendicular to the radius there. A secant passes through the circle at two distinct points. These definitions are mutually exclusive—a line cannot simultaneously touch at one point and intersect at two points. These 3-mark questions blend visual proof with numerical substitution. Success requires a clear diagram and systematic use of congruence or Pythagoras.

5-mark questions demand complete logical scaffolding, detailed diagrams, and often chain multiple sub-parts. Here are three high-confidence patterns: **Q1: Prove that the tangent at any point on a circle is perpendicular to the radius at that point of contact.** Full Solution: — Given: A circle with centre O. A tangent line l touches the circle at point P. — To Prove: OP ⊥ l (i.e., angle between OP and tangent = 90°). — Proof: Assume tangent l is not perpendicular to OP. Then we can draw a perpendicular from O to line l, meeting it at point Q. Since Q lies on l and OQ ⊥ l, the distance OQ < OP. But P is the point of contact on the circle, so OP = radius. This means Q is inside the circle. Yet l is supposed to be a tangent (touching at only one point), so no other point of l can be inside or on the circle. This contradiction shows our assumption is wrong. Therefore, l ⊥ OP at P. — Diagram: Draw circle, centre O, point P on circumference, tangent l at P, and OP meeting l at 90°. — Award marks: Clear statement of given/to prove (1 mark), logical flow of proof (2 marks), diagram (1 mark), conclusion (1 mark). **Q2: Two tangents PA and PB are drawn from an external point P to a circle with centre O and radius 5 cm. The angle APB = 70°. Find: (a) angle AOB, (b) angle OAP, (c) length PA if OP = 15 cm.** Full Solution: (a) In quadrilateral OAPB: angle OAP = 90° (radius ⊥ tangent), angle OBP = 90°, angle APB = 70°. Sum = 360°, so angle AOB = 360° − 90° − 90° − 70° = 110°. (1.5 marks) (b) Since PA is tangent and OA is radius, angle OAP = 90°. (0.5 marks) (c) In right triangle OAP: OA = 5, OP = 15. By Pythagoras, PA² = OP² − OA² = 225 − 25 = 200, so PA = 10√2 cm ≈ 14.14 cm. (1.5 marks) **Q3: From an external point P, two tangents PA and PB are drawn to a circle with centre O. If angle OPA = 35°, find angles APB, AOB, and OBP. Prove that PA = PB.** Full Solution: — In triangle OPA: angle OAP = 90° (tangent ⊥ radius), angle OPA = 35°. So angle AOP = 180° − 90° − 35° = 55°. — By symmetry (or by congruent triangles OPA ≅ OPB via RHS), angle OPB = 35°, angle BOP = 55°, angle OBP = 90°. — Therefore, angle APB = angle OPA + angle OPB = 35° + 35° = 70°. (1 mark) — angle AOB = angle AOP + angle BOP = 55° + 55° = 110°. (1 mark) — angle OBP = 90°. (0.5 marks) — Proof of PA = PB: In triangles OPA and OPB, OA = OB (radii), angle OAP = angle OBP = 90°, OP = OP (common). By RHS, △OPA ≅ △OPB, so PA = PB. (2.5 marks) Each 5-mark solution requires a sketch, labelled angles, and explicit citation of geometric axioms (Pythagoras, congruence rules, angle sums). Examiners value step clarity over speed.

The 2024–25 rationalized CBSE syllabus preserved Chapter 10 core content, but examiner focus has subtly shifted. Here's what to watch: First, case-based questions are rising. Instead of 'prove tangent ⊥ radius,' expect 'a telephone pole 8 m high needs guy-wires as tangents from ground to stabilize it; if the pole's base is 6 m from an observation point, find wire length.' These demand proof + contextual application. Second, diagram construction is weighted more heavily. The 5-mark proof questions now expect you not just to write logic, but to draw precise circles, mark radii, tangents, and angles. Sloppy sketches lose marks even if math is correct. Third, angle-chasing combinations are more complex. Older papers tested single 3-mark angles. Current papers ask: 'tangent PA, angle APB = 60°, find angle AOB, then use it to find arc AB.' This chains concepts. Fourth, Pythagoras substitutions are more implicit. Papers no longer hand you 'OT = 5, PT = 12, find OP.' Now it's 'a circle is inscribed in a right triangle; tangent touches the triangle's sides; find the inradius given side lengths.' Fifth, coordinate geometry might merge in. While Chapter 10 is traditionally synthetic, newer papers sometimes ask 'find the tangent equation to circle x² + y² = 25 at point (3, 4).' Check your mock papers to see if your school includes this. Recommendation: practice mixed-topic papers and case-based scenarios alongside pure tangent proofs. Cbsetutor.ai's adaptive quizzes update with each year's paper, so you stay ahead of emerging patterns.

When the exam paper lands on your desk, here's your chapter-wise approach for circles: Read the full paper first (2 minutes). Identify all Chapter 10 questions. Note their marks: 1-mark, 3-mark, or 5-mark. Sketch the chapter's geometry framework on scrap: circle, tangent, radius, right angle, congruence rules. This primes your brain. Tackle 1-mark questions first (3–4 minutes total). These are high-confidence marks; don't lose them to careless errors. For each, redraw the geometry if needed—a wrong mental picture leads to wrong answers. Do 3-mark questions next (6–8 minutes per question). Write down 'Given:' and 'To Find:' before jumping in. Use a ruler for diagrams. Check: Does my tangent actually touch once? Is my radius truly perpendicular? Did I apply the congruence rule name (SSS, SAS, RHS, etc.)? Reserve 5-mark questions for last (10–12 minutes per question). These carry heavy marks and often have multiple sub-parts. Read all parts before solving—sometimes earlier parts set up later ones. Label your diagram extensively: O for centre, P for external point, A and B for tangent touch points. Restate the congruence criterion explicitly: 'By RHS congruence, triangle OAP ≅ triangle OBP.' Write 'Therefore,' before conclusions. If stuck on a 5-mark proof, write what you know (angle-angle pairs, equal radii, perpendicular tangent) and let the examiner award partial marks for valid reasoning. Don't erase unless certain—rough work showing thought process often earns credit. Double-check Pythagoras arithmetic: OT² + PT² = OP². Use units consistently (cm, m, units²). If a tangent-length question gives OP = 13 and radius = 5, mentally verify: 5² + 12² = 13²? Yes. So PT = 12. This self-check saves errors. Finally, allocate your last 2 minutes to skim Chapter 10 answers: is my angle sum = 360° for the quadrilateral? Did I write 'tangent ⊥ radius' where needed? A quick visual scan catches careless lapses.

This page consolidates real CBSE question patterns so you focus your study. Here's the workflow: Week 1: Read the 1-mark section. For each of the five questions, close this page and write the answer from memory. Check. Repeat until three in a row are correct. Week 2: Move to 3-mark questions. Solve each without looking at the solution. Time yourself (5 minutes per question). Compare your proof or calculation against the answer. Did you name the triangles? Did you cite the congruence rule? Adjust your writing. Week 3: Attempt 5-mark questions. These are full proofs. Solve on paper (not here). Draw a circle with a ruler. Label clearly. Write proofs in complete sentences, not shorthand. Only then compare. Week 4: Solve mixed tests. Take a mock paper or use past years from your school, and attempt all Chapter 10 questions under exam conditions (time limit, no calculator for most, no looking at answers mid-attempt). Scan your answers against the solutions here. Identify patterns in what you got wrong: careless arithmetic, proof logic gaps, or diagram confusion. Retarget weak areas. Before your board exam, do a final 30-minute sprint: solve the three 5-mark questions again, fresh, to build muscle memory for proof structure. If you want guided feedback on your proofs or adaptive problem sets that adjust to your level, start a 3-day free trial at cbsetutor.ai. Our AI tutor grades your proof step-by-step, pinpoints exactly where your logic breaks, and gives you harder follow-ups if you're nailing it. Pair self-study from this page with interactive practice for the fastest score lift.