{
  "title": "Proof_that_radius_is_perpendicular_to_tangent(Circles) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "The angle between radius and tangent at point of contact is:", "options": {"A": "30°", "B": "45°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "A line intersecting circle at exactly one point is called:", "options": {"A": "Secant", "B": "Tangent", "C": "Chord", "D": "Diameter"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "From point P, if PT is tangent to circle with center O, then ∠OTP = ?", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "The point where tangent touches circle is called:", "options": {"A": "Center", "B": "Point of contact", "C": "Midpoint", "D": "Endpoint"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "How many tangents can be drawn from point on circle?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If radius is perpendicular to line at point of contact, then line is:", "options": {"A": "Chord", "B": "Secant", "C": "Tangent", "D": "Diameter"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "In right triangle formed by radius, tangent and line from center to external point, hypotenuse is:", "options": {"A": "Radius", "B": "Tangent", "C": "Line from center to external point", "D": "None"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "If PT is tangent and OT is radius, then PT² + OT² = ?", "options": {"A": "OP", "B": "OP²", "C": "2OP", "D": "2OP²"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Tangents drawn from external point are:", "options": {"A": "Equal", "B": "Unequal", "C": "Parallel", "D": "Perpendicular"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If ∠OTP = 90°, then PT is:", "options": {"A": "Chord", "B": "Tangent", "C": "Secant", "D": "Diameter"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "In circle with center O, PT is tangent at T. If OT = 5 cm, OP = 13 cm, find PT.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "The longest chord of circle is:", "options": {"A": "Radius", "B": "Tangent", "C": "Diameter", "D": "Secant"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "Angle in semicircle is:", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "If line touches circle at one point only, it is:", "options": {"A": "Tangent", "B": "Secant", "C": "Chord", "D": "Diameter"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "From external point, number of tangents to circle is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "In ∆OTP right angled at T, if OT=3, OP=5, find PT.", "options": {"A": "2", "B": "3", "C": "4", "D": "6"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "Tangent is perpendicular to radius at:", "options": {"A": "Center", "B": "Any point", "C": "Point of contact", "D": "Midpoint"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "If PT is tangent and OT is radius, which angle is 90°?", "options": {"A": "∠POT", "B": "∠OTP", "C": "∠OPT", "D": "∠PTO"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "A line perpendicular to radius at outer end is:", "options": {"A": "Chord", "B": "Tangent", "C": "Secant", "D": "Diameter"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "In circle, shortest distance from center to tangent is:", "options": {"A": "0", "B": "Radius", "C": "Diameter", "D": "Chord"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Prove that tangent at any point of circle is perpendicular to radius through point of contact.", "options": {"A": "By contradiction method", "B": "Using Pythagoras theorem", "C": "Using congruent triangles", "D": "All of above"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "In circle with center O, PT is tangent at T. If OP = 10 cm, PT = 8 cm, find radius.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "From point P, two tangents PA and PB are drawn to circle with center O. If ∠APB = 50°, find ∠AOB.", "options": {"A": "100°", "B": "115°", "C": "130°", "D": "145°"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "In figure, if TP and TQ are tangents to circle with center O so that ∠POQ = 110°, find ∠PTQ.", "options": {"A": "55°", "B": "60°", "C": "65°", "D": "70°"}, "correct_answer": "D"},
    {"id": 25, "difficulty": "medium", "question": "If tangents PA and PB from point P to circle with center O are inclined at 80°, find ∠POA.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "Prove that lengths of tangents from external point are equal. What theorem proves this?", "options": {"A": "SSS congruence", "B": "SAS congruence", "C": "RHS congruence", "D": "ASA congruence"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "In circle of radius 5 cm, tangent PT is drawn from point P such that PT = 12 cm. Find OP.", "options": {"A": "11 cm", "B": "12 cm", "C": "13 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "If PA and PB are tangents from P to circle with center O, and ∠APB = 60°, find ∠OAP.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "Prove that line drawn through end point of radius and perpendicular to it is tangent to circle.", "options": {"A": "True statement", "B": "False statement", "C": "Sometimes true", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Two concentric circles have radii 5 cm and 3 cm. Find length of chord of larger circle which touches smaller circle.", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "In quadrilateral ABCD circumscribing circle, prove that AB + CD = AD + BC.", "options": {"A": "Using tangent properties", "B": "Using triangle properties", "C": "Using circle theorems", "D": "All of above"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If OP = diameter, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "In circle with center O, tangent XY touches circle at P. If OQ ⊥ XY, where Q is on XY, then:", "options": {"A": "OQ > OP", "B": "OQ = OP", "C": "OQ < OP", "D": "OQ = 2OP"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "If TP and TQ are tangents from T to circle with center O, and ∠PTQ = 70°, find ∠OPQ.", "options": {"A": "20°", "B": "25°", "C": "30°", "D": "35°"}, "correct_answer": "D"},
    {"id": 35, "difficulty": "medium", "question": "Prove that angle between two tangents drawn from external point is supplementary to angle subtended by line segments joining points of contact to center.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "In figure, if ∠ATO = 40°, where AT is tangent and OA is radius, find ∠AOT.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "Two tangents TP and TQ are drawn from T to circle with center O. If ∠PTQ = 60° and radius = 7 cm, find PT.", "options": {"A": "7 cm", "B": "7√3 cm", "C": "14 cm", "D": "14√3 cm"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "In circle, chord of length 24 cm is at distance 5 cm from center. Find radius of circle tangent to this chord at its midpoint.", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Prove that parallelogram circumscribing circle is rhombus.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If angle between two tangents drawn from point P to circle of radius 'a' and center O is 60°, find OP.", "options": {"A": "a", "B": "2a", "C": "a√2", "D": "a√3"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "In circle with center O, tangent AB touches circle at B. If OA = 13 cm, AB = 12 cm, find radius.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "Two concentric circles have radii 13 cm and 5 cm. Find length of chord of larger circle that touches smaller circle.", "options": {"A": "12 cm", "B": "18 cm", "C": "24 cm", "D": "30 cm"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "medium", "question": "Prove that tangents drawn at ends of diameter are parallel.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "In quadrilateral ABCD circumscribing circle, if AB = 6 cm, BC = 7 cm, CD = 4 cm, find AD.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "From point A, tangents AB and AC are drawn to circle with center O. If ∠BAC = 120°, find ∠BOC.", "options": {"A": "100°", "B": "110°", "C": "120°", "D": "130°"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "In ∆ABC right angled at B, circle inscribed touches AB at D. If AD = 5 cm, DC = 7 cm, find radius.", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Prove that angle between tangent and chord through point of contact equals angle in alternate segment.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Two circles touch externally at P. Common tangent touches circles at A and B. Prove that ∠APB = 90°.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "In circle with center O, tangent PQ touches circle at Q. If OQ = 6 cm, OP = 10 cm, find area ∆OPQ.", "options": {"A": "24 cm²", "B": "30 cm²", "C": "36 cm²", "D": "48 cm²"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "If PA and PB are tangents from P to circle with center O, and radius = 8 cm, OP = 17 cm, find perimeter ∆PAB.", "options": {"A": "30 cm", "B": "34 cm", "C": "38 cm", "D": "42 cm"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Prove that opposite sides of quadrilateral circumscribing circle subtend supplementary angles at center.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "In figure, if ∠AOB = 125°, find ∠OAB where A and B are points of contact of tangents from O.", "options": {"A": "27.5°", "B": "32.5°", "C": "35°", "D": "37.5°"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Two circles intersect at A and B. From point P on AB produced, tangents PT and PS are drawn to circles. Prove PT = PS.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "In ∆ABC, AB = 8 cm, BC = 10 cm, CA = 12 cm. Circle touches BC at D, CA at E, AB at F. Find BD.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Prove that line joining center to point of intersection of two tangents bisects angle between tangents.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If OP = 2r, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 57, "difficulty": "medium", "question": "Two tangents TP and TQ are drawn from T to circle with center O. If ∠PTQ = 80°, find ∠POQ.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "medium", "question": "Prove that angle between tangent and radius at point of contact is 90° using method of contradiction.", "options": {"A": "Valid proof", "B": "Invalid proof", "C": "Partial proof", "D": "No proof"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "In circle with center O, tangent PQ touches circle at Q. PR is chord. If ∠RPQ = 40°, find ∠ROQ.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "If PA and PB are tangents from P to circle with center O, and ∠APB = 50°, find ∠OAB where A and B are points of contact.", "options": {"A": "25°", "B": "30°", "C": "35°", "D": "40°"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "Prove that lengths of tangents from external point are equal. Which triangles are congruent?", "options": {"A": "∆OAP ≅ ∆OBP", "B": "∆OAB ≅ ∆OPA", "C": "∆OAP ≅ ∆OBP by RHS", "D": "∆OAP ≅ ∆OBP by RHS"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "In quadrilateral ABCD circumscribing circle, if AB = 5 cm, BC = 7 cm, CD = 9 cm, find AD.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "Two circles touch internally at A. Tangent to smaller circle at B cuts larger circle at C and D. Prove AB bisects ∠CAD.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "From point P, 25 cm from center O of circle of radius 7 cm, tangents PA and PB are drawn. Find perimeter quadrilateral OAPB.", "options": {"A": "56 cm", "B": "60 cm", "C": "64 cm", "D": "68 cm"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Prove that tangents at ends of chord make equal angles with chord.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "In circle with center O, tangent XY touches circle at P. Chord PQ is drawn. If ∠XPQ = 50°, find ∠POQ.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "Two concentric circles have radii r and R (r < R). Chord of larger circle touches smaller circle. Find length of chord.", "options": {"A": "2√(R² - r²)", "B": "√(R² - r²)", "C": "2√(R² + r²)", "D": "√(R² + r²)"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Prove that angle between two tangents from external point is supplementary to angle subtended by chord of contact at center.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "In ∆ABC, circle inscribed touches BC at D. If BD = 4 cm, DC = 6 cm, find AB - AC.", "options": {"A": "0 cm", "B": "1 cm", "C": "2 cm", "D": "3 cm"}, "correct_answer": "C"},
    {"id": 70, "difficulty": "medium", "question": "From point P outside circle, tangents PA and PB are drawn. If PA = 8 cm, OP = 10 cm, find radius.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "Prove that quadrilateral formed by angle bisectors of cyclic quadrilateral is also cyclic.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "In circle with center O, tangent PT touches circle at T. If PT = 12 cm, PO = 13 cm, find area quadrilateral OPTQ where TQ is diameter.", "options": {"A": "60 cm²", "B": "65 cm²", "C": "70 cm²", "D": "75 cm²"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "Two circles intersect at A and B. Tangents at A to circles meet circles again at C and D. Prove ∠CBD = 2∠CAD.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "In ∆ABC, AB = 12 cm, BC = 8 cm, CA = 10 cm. Circle inscribed touches AB at D. Find AD.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "C"},
    {"id": 75, "difficulty": "medium", "question": "Prove that line drawn from center perpendicular to chord bisects chord.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If ∠OAB = 30°, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 77, "difficulty": "medium", "question": "Two circles touch externally at C. Common tangent AB touches circles at A and B. Prove AC ⊥ BC.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "In quadrilateral ABCD circumscribing circle, if AD = 23 cm, AB + CD = 35 cm, find BC.", "options": {"A": "11 cm", "B": "12 cm", "C": "13 cm", "D": "14 cm"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "Prove that angle between tangent and chord equals angle in alternate segment. This is called:", "options": {"A": "Tangent-chord theorem", "B": "Alternate segment theorem", "C": "Chord theorem", "D": "Tangent theorem"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "From point P, 17 cm from center O of circle of radius 8 cm, tangents PA and PB are drawn. Find area ∆PAB.", "options": {"A": "240/17 cm²", "B": "480/17 cm²", "C": "720/17 cm²", "D": "960/17 cm²"}, "correct_answer": "D"},
    {"id": 81, "difficulty": "hard", "question": "Prove that radius is perpendicular to tangent at point of contact using method of contradiction. Assume it's not perpendicular and show contradiction.", "options": {"A": "Valid proof", "B": "Invalid proof", "C": "Incomplete proof", "D": "Circular reasoning"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Two circles intersect at A and B. Through A, line is drawn meeting circles at C and D. Tangents at C and D intersect at P. Prove P, C, B, D concyclic.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "In ∆ABC, incircle touches BC, CA, AB at D, E, F. If AB = 14 cm, BC = 15 cm, CA = 13 cm, find BD, CE, AF.", "options": {"A": "BD=7, CE=6, AF=8", "B": "BD=8, CE=5, AF=7", "C": "BD=7, CE=5, AF=8", "D": "BD=8, CE=6, AF=7"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "Prove that angle between two tangents from external point is supplementary to angle subtended by chord of contact at center. For circle radius 5 cm, OP=13 cm, verify.", "options": {"A": "∠APB=67.4°, ∠AOB=112.6°", "B": "∠APB=67.4°, ∠AOB=112.6°", "C": "∠APB=67.4°, ∠AOB=112.6°", "D": "∠APB=67.4°, ∠AOB=112.6°"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "From point P outside circle, tangents PA and PB are drawn. Chord AC is drawn parallel to PB. Prove PC bisects ∠APB.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Two circles touch internally at A. Chord BC of larger circle touches smaller circle at D. Prove AD bisects ∠BAC.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Prove that parallelogram circumscribing circle is rhombus. If sides are 15 cm and 20 cm, find radius.", "options": {"A": "6 cm", "B": "7 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "In quadrilateral ABCD, incircle touches sides at P, Q, R, S. Prove that circles with diameters AC and BD are orthogonal.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Prove that if two tangents are drawn from external point, then line joining center to external point bisects angle between tangents and also bisects angle between radii to points of contact.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Two circles intersect at A and B. Tangents at A to circles meet circles again at C and D. Prove AB² = BC × BD.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "In ∆ABC, incircle touches BC at D. Circle with diameter BC touches incircle. Prove AB = AC.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Prove that radical axis of two circles is perpendicular to line joining centers. For circles x²+y²=25 and (x-8)²+y²=9, find radical axis.", "options": {"A": "x=5", "B": "x=6", "C": "x=7", "D": "x=8"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "From point P, tangents PA and PB are drawn to circle with center O. Through O, line parallel to PA is drawn meeting PB at Q. Prove that Q is midpoint PB.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Two circles intersect at A and B. Through A, line is drawn meeting circles at C and D. Tangents at C and D intersect at P. Prove ∠CPD + ∠CBD = 180°.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Prove that if from any point on common chord of two intersecting circles, tangents are drawn to circles, then they are equal.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "In quadrilateral ABCD circumscribing circle, diagonals AC and BD intersect at O. Prove that OA × OC = OB × OD.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Prove that angle between tangent and chord through point of contact equals angle in alternate segment. Use this to prove that if from point of contact chord is drawn, then angles in alternate segments are equal.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Two circles touch internally at A. Chord BC of larger circle touches smaller circle at D. Prove that AD produced bisects arc BC of larger circle not containing A.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Prove that if two circles intersect, line joining centers is perpendicular bisector of common chord. For circles with centers (0,0) radius 5 and (8,0) radius 3, verify.", "options": {"A": "Common chord: x=5, length=6", "B": "Common chord: x=5, length=6", "C": "Common chord: x=5, length=6", "D": "Common chord: x=5, length=6"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Prove that tangents drawn at ends of diameter are parallel. Using coordinate geometry, for circle x²+y²=r², show tangents at (r,0) and (-r,0) are parallel.", "options": {"A": "Both tangents: x=r and x=-r", "B": "Both vertical lines", "C": "Both parallel to y-axis", "D": "All of above"}, "correct_answer": "D"}
  ]
}