{
  "title": "Tangent to a Circle - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "A line that intersects a circle in exactly one point is called:", "options": {"A": "Chord", "B": "Secant", "C": "Tangent", "D": "Diameter"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "How many tangents can be drawn to a circle through a point on the circle?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "At the point of contact, the tangent is perpendicular to:", "options": {"A": "Chord", "B": "Diameter", "C": "Radius", "D": "Secant"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If a line touches a circle and is perpendicular to the radius at the point of contact, then it is:", "options": {"A": "Secant", "B": "Chord", "C": "Tangent", "D": "Diameter"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "From an external point, the number of tangents to a circle is:", "options": {"A": "1", "B": "2", "C": "3", "D": "0"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "The common point of a tangent and a circle is called:", "options": {"A": "Center", "B": "Midpoint", "C": "Point of contact", "D": "Vertex"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "If two tangents are drawn from an external point to a circle, they are:", "options": {"A": "Parallel", "B": "Perpendicular", "C": "Equal in length", "D": "Unequal"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "In a circle with center O, if PT is tangent at T, then ∠OTP is:", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "A line drawn through the end point of a radius and perpendicular to it is:", "options": {"A": "Chord", "B": "Tangent", "C": "Secant", "D": "Diameter"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If a circle touches all sides of a quadrilateral, then the sum of opposite sides is:", "options": {"A": "Equal", "B": "Unequal", "C": "Double", "D": "Half"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "The angle between tangent and chord through point of contact is equal to angle in:", "options": {"A": "Same segment", "B": "Alternate segment", "C": "Opposite segment", "D": "Adjacent segment"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If PA and PB are tangents from P to circle with center O, then triangle OAP is:", "options": {"A": "Isosceles", "B": "Right angled", "C": "Equilateral", "D": "Scalene"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "A line intersecting circle in two points is:", "options": {"A": "Tangent", "B": "Secant", "C": "Radius", "D": "Diameter"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "From a point inside circle, number of tangents that can be drawn is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "In cyclic quadrilateral, sum of opposite angles is:", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If two circles touch externally, number of common tangents is:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "If two circles touch internally, number of common tangents is:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "The length of tangent from point P to circle with center O and radius r, if OP = d, is:", "options": {"A": "√(d² + r²)", "B": "√(d² - r²)", "C": "√(r² - d²)", "D": "d² - r²"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If two concentric circles have radii 5 cm and 3 cm, length of chord of larger circle touching smaller circle is:", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "In circle with center O, PT is tangent. If ∠TPO = 30°, then ∠TOP is:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},

    {"id": 21, "difficulty": "medium", "question": "From point P, 25 cm from center O of circle of radius 7 cm, length of tangent is:", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "Two circles of radii 5 cm and 3 cm touch externally. Distance between centers is:", "options": {"A": "2 cm", "B": "5 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "In circle with center O, PT is tangent at T. Chord TR makes 35° with tangent. ∠TOR is:", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "Two tangents PA and PB from point P to circle make 70° angle. ∠AOB where O is center is:", "options": {"A": "70°", "B": "110°", "C": "140°", "D": "160°"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Chord AB of circle with center O is 16 cm. OM ⟂ AB, OM = 6 cm. Length of tangent from A if OA = 10 cm is:", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "Two circles of radii 8 cm and 3 cm have centers 13 cm apart. Length of direct common tangent is:", "options": {"A": "12 cm", "B": "√120 cm", "C": "√133 cm", "D": "√144 cm"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "In circle, chord AB = 24 cm, distance from center to AB is 5 cm. Length of tangent from A if OA = 13 cm is:", "options": {"A": "5 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "From point P, tangent PT = 12 cm, secant PAB with PA = 8 cm. AB is:", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "Two circles touch internally. Larger radius 10 cm, distance between centers 6 cm. Smaller radius is:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "PA and PB are tangents from P to circle with center O. If ∠APB = 50° and radius = 7 cm, length of tangent is:", "options": {"A": "7 tan 25°", "B": "7 cot 25°", "C": "7 sin 25°", "D": "7 cos 25°"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "Two circles intersect at A and B. Their radii 10 cm and 8 cm, centers 12 cm apart. AB is:", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "D"},
    {"id": 32, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR when produced meets tangent at P. If ∠PTR = 30°, ∠RPT = 50°, ∠TRP is:", "options": {"A": "50°", "B": "60°", "C": "80°", "D": "100°"}, "correct_answer": "D"},
    {"id": 33, "difficulty": "medium", "question": "Two circles of radii 5 cm and 12 cm have transverse common tangent 15 cm. Distance between centers:", "options": {"A": "13 cm", "B": "17 cm", "C": "20 cm", "D": "25 cm"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn. PA = 8 cm, ∠APB = 60°, OP = 10 cm. Radius is:", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Two circles touch externally. Radii ratio 3:2, distance between centers 15 cm. Radii are:", "options": {"A": "6 cm, 9 cm", "B": "8 cm, 12 cm", "C": "9 cm, 6 cm", "D": "12 cm, 8 cm"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "In circle with center O, AB is diameter. Tangent at B meets AC produced at T. If ∠CAB = 40°, ∠BTC is:", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "Two concentric circles radii 13 cm and 5 cm. Chord of larger circle touching smaller circle has length:", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "From point A, tangents AB and AC to circle. BC = 8 cm, radius = 3 cm. AB is:", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Two circles radii 7 cm and 1 cm touch externally. Direct common tangent length:", "options": {"A": "4√3 cm", "B": "4√5 cm", "C": "4√6 cm", "D": "4√7 cm"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "In circle with center O, PA and PB are tangents. If ∠OAB = 25°, ∠APB is:", "options": {"A": "25°", "B": "50°", "C": "65°", "D": "70°"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "Two circles intersect at A and B. ∠OAO' = 40°, ∠OBO' = 70° where O, O' are centers. Angle between tangents at A:", "options": {"A": "30°", "B": "40°", "C": "50°", "D": "60°"}, "correct_answer": "D"},
    {"id": 42, "difficulty": "medium", "question": "Chord AB = 30 cm, distance from center to AB = 8 cm. Tangent from A if OA = 17 cm:", "options": {"A": "8 cm", "B": "15 cm", "C": "17 cm", "D": "20 cm"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "Two circles radii 8 cm and 3 cm, centers 13 cm apart. Common chord length when they intersect:", "options": {"A": "48/13 cm", "B": "96/13 cm", "C": "120/13 cm", "D": "144/13 cm"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "From point P, tangent PT = 10 cm, secant PAB with PA = 6 cm. AB is:", "options": {"A": "14/3 cm", "B": "20/3 cm", "C": "32/3 cm", "D": "40/3 cm"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "Two circles touch internally. Larger radius 12 cm, distance between centers 8 cm. Smaller radius:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "In circle, PA is tangent. Chord AB makes 40° with tangent. Angle subtended by AB at center:", "options": {"A": "40°", "B": "80°", "C": "100°", "D": "140°"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "Two circles radii 10 cm and 6 cm, centers 20 cm apart. Direct common tangent length:", "options": {"A": "12 cm", "B": "16 cm", "C": "√336 cm", "D": "√384 cm"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "From point P outside circle, PA is tangent, PBC is secant. PA = 6 cm, PB = 2 cm. PC is:", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "18 cm"}, "correct_answer": "D"},
    {"id": 49, "difficulty": "medium", "question": "Two circles touch externally. Sum of areas 130π cm², centers 14 cm apart. Radii if differ by 2 cm:", "options": {"A": "5 cm, 7 cm", "B": "6 cm, 8 cm", "C": "7 cm, 9 cm", "D": "8 cm, 10 cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "In circle with center O, PA and PB are tangents. Line through P parallel to AB meets OA at C. OA = 5 cm, OP = 13 cm. Area ratio ΔOAP:ΔOCP:", "options": {"A": "169:144", "B": "144:25", "C": "169:25", "D": "25:144"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Two circles of radii 5 cm and 3 cm intersect. Common chord 4 cm. Distance between centers:", "options": {"A": "2√5 cm", "B": "3√3 cm", "C": "4√2 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "From point A, tangents AB and AC to circle. If BC = 10 cm and radius = 5 cm, AB is:", "options": {"A": "5√3 cm", "B": "10 cm", "C": "10√2 cm", "D": "12 cm"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Two circles radii 13 cm and 5 cm have direct common tangent 24 cm. Distance between centers:", "options": {"A": "26 cm", "B": "28 cm", "C": "30 cm", "D": "32 cm"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "In circle, chord AB = 24 cm, distance from center to AB = 5 cm. Tangent from B if OB = 13 cm:", "options": {"A": "5 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "Two circles touch internally. Sum of radii 18 cm, difference 8 cm. Distance between centers:", "options": {"A": "5 cm", "B": "8 cm", "C": "10 cm", "D": "13 cm"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR subtends 80° at center. ∠PTR is:", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Two concentric circles radii 17 cm and 8 cm. Chord of larger circle touching smaller circle:", "options": {"A": "30 cm", "B": "32 cm", "C": "34 cm", "D": "36 cm"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "From point P, tangents PA and PB to circle. Third tangent touches at C, intersects PA and PB at Q and R. Perimeter ΔPQR = 24 cm. PA is:", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "medium", "question": "Two circles radii 5 cm and 12 cm intersect orthogonally. Distance between centers:", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "19 cm"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "In circle with center O, PA and PB are tangents. OP is diameter of circle through A and B. Radius first circle = 6 cm. OP is:", "options": {"A": "6√2 cm", "B": "12 cm", "C": "12√2 cm", "D": "18 cm"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "Two circles of radii 7 cm and 1 cm have direct common tangent AB (A on larger). Distance from A to point where line joining centers meets tangent if centers 10 cm apart:", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "From point P, tangents PA and PB to circle. Chord AC ∥ PB. PA = 8 cm, radius = 5 cm. BC is:", "options": {"A": "6.4 cm", "B": "8 cm", "C": "9.6 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "medium", "question": "Three equal circles touch externally. Ratio area triangle formed by centers to area sector between common tangents:", "options": {"A": "√3:π", "B": "2√3:π", "C": "3√3:π", "D": "4√3:π"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Two circles touch externally at P. Common tangent touches at A and B. Another tangent from A to second circle touches at C. AB = 6 cm. AC is:", "options": {"A": "3 cm", "B": "3√2 cm", "C": "4 cm", "D": "4√2 cm"}, "correct_answer": "B"},
    {"id": 65, "difficulty": "medium", "question": "In circle with center O, AB is chord. Tangents at A and B intersect at P. ∠APB = 50°. ∠AOB is:", "options": {"A": "115°", "B": "125°", "C": "130°", "D": "140°"}, "correct_answer": "C"},
    {"id": 66, "difficulty": "medium", "question": "Two circles radii 5 cm and 12 cm, direct common tangent 15 cm. Distance between centers if on same side of tangent:", "options": {"A": "√106 cm", "B": "√154 cm", "C": "√194 cm", "D": "√234 cm"}, "correct_answer": "C"},
    {"id": 67, "difficulty": "medium", "question": "From point P outside circle, tangents PA and PB. Through midpoint M of AB, line PM meets circle at C and D. PM = 9 cm, MC = 4 cm. Radius:", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "Two circles touch internally at A. Chord BC of larger circle touches smaller at D. ∠BAC = 60°. Ratio BD:DC:", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "Two circles intersect orthogonally. Radii 6 cm and 8 cm. Common chord length:", "options": {"A": "48/5 cm", "B": "96/5 cm", "C": "24/5 cm", "D": "12/5 cm"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "In circle with center O, PA and PB are tangents. Line through P ∥ AB meets OA at C, OB at D. PA = 12 cm, radius = 5 cm. Area trapezium ABDC:", "options": {"A": "60 cm²", "B": "120 cm²", "C": "180 cm²", "D": "240 cm²"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "Three circles radii 3 cm, 4 cm, 5 cm touch externally. Area triangle formed by centers:", "options": {"A": "12 cm²", "B": "18 cm²", "C": "24 cm²", "D": "36 cm²"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "medium", "question": "In circle with center O, AB is diameter. From T on tangent at B, TA meets circle at C. ∠BTA = 20°. ∠BOC is:", "options": {"A": "40°", "B": "60°", "C": "80°", "D": "100°"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "medium", "question": "Two circles intersect at A and B. Through A, line PAQ meets circles at P and Q. Tangents at P and Q intersect at T. If ∠APB = 50°, ∠AQB = 60°, ∠PTQ is:", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "In circle with center O, PA and PB are tangents. Line through P ∥ AB meets OA at C. OA = 5 cm, OP = 13 cm. CD where D on OB:", "options": {"A": "60/13 cm", "B": "120/13 cm", "C": "130/13 cm", "D": "150/13 cm"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "Two circles radii 5 cm and 12 cm intersect. Common chord 8 cm. Distance between centers:", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "19 cm"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "From point P, tangents PA and PB to circle. If ∠APB = 70° and radius = 8 cm, length of tangent:", "options": {"A": "8 tan 35°", "B": "8 cot 35°", "C": "8 sin 35°", "D": "8 cos 35°"}, "correct_answer": "B"},
    {"id": 77, "difficulty": "medium", "question": "Two circles radii 8 cm and 3 cm, centers 10 cm apart. Transverse common tangent length:", "options": {"A": "6 cm", "B": "8 cm", "C": "√84 cm", "D": "√96 cm"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "medium", "question": "In circle, PQ is diameter. Tangent at Q meets PR at T where R on circle. ∠QPR = 40°. ∠QTR is:", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "Two circles touch internally. Radius larger 9 cm, distance between centers 5 cm. Smaller radius:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 80, "difficulty": "medium", "question": "From point A, tangents AB and AC to circle. ∠BAC = 50°. ∠BOC where O is center:", "options": {"A": "100°", "B": "115°", "C": "130°", "D": "140°"}, "correct_answer": "C"},

    {"id": 81, "difficulty": "hard", "question": "Two circles with centers O and O' intersect at A and B. Tangents at A to circles intersect at P. If ∠OAO' = 50° and ∠OBO' = 80°, find ∠APB.", "options": {"A": "30°", "B": "40°", "C": "50°", "D": "60°"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "From external point P, tangents PA and PB to circle with center O. Another circle with OP as diameter passes through A and B. If radius of first circle is 8 cm, find area of quadrilateral OAPB.", "options": {"A": "64 cm²", "B": "96 cm²", "C": "128 cm²", "D": "192 cm²"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "Three circles with radii 6 cm, 7 cm, and 8 cm touch each other externally. Find area of triangle formed by joining centers.", "options": {"A": "12√5 cm²", "B": "21√6 cm²", "C": "24√7 cm²", "D": "36√2 cm²"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "In circle with center O, PA and PB are tangents. Line through P parallel to AB meets OA at C and OB at D. If PA = 15 cm and radius = 8 cm, find ratio of perimeters of triangles OAP and OCD.", "options": {"A": "15:8", "B": "17:8", "C": "15:7", "D": "17:7"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "Two circles intersect orthogonally. Product of their radii is 144 and distance between centers is 13 cm. Find sum of radii.", "options": {"A": "17 cm", "B": "19 cm", "C": "21 cm", "D": "23 cm"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "From point P, tangents PA and PB to circle. A third tangent touches circle at C and intersects PA at Q, PB at R. If PQ = 4 cm, QR = 5 cm, RP = 6 cm, find radius of circle.", "options": {"A": "√15 cm", "B": "√21 cm", "C": "√24 cm", "D": "√30 cm"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "Two circles of radii 15 cm and 8 cm have direct common tangent of length 20 cm. Find distance between centers if circles are on opposite sides of tangent.", "options": {"A": "21 cm", "B": "23 cm", "C": "25 cm", "D": "27 cm"}, "correct_answer": "C"},
    {"id": 88, "difficulty": "hard", "question": "In circle with center O, chord AB subtends 120° at O. Tangents at A and B intersect at P. Find ratio of area of triangle APB to area of sector OAB.", "options": {"A": "3√3:π", "B": "4√3:π", "C": "6√3:π", "D": "8√3:π"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "Two circles touch internally at A. Chord BC of larger circle is tangent to smaller circle at D. If BC = 24 cm and distance from A to BC is 10 cm, find ratio of radii.", "options": {"A": "3:2", "B": "5:3", "C": "7:4", "D": "9:5"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "Three circles of equal radii r touch each other externally. Find area of region between circles (curvilinear triangle).", "options": {"A": "r²(√3 - π/2)", "B": "r²(2√3 - π)", "C": "r²(√3 - π/3)", "D": "r²(3√3 - π)"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "From point P outside circle, PA and PB are tangents. Chord AC is drawn parallel to PB meeting circle at C. If PA = 12 cm and radius = 7.5 cm, find length of BC.", "options": {"A": "9 cm", "B": "12 cm", "C": "15 cm", "D": "18 cm"}, "correct_answer": "C"},
    {"id": 92, "difficulty": "hard", "question": "Two circles intersect at A and B. Through A, lines PAC and QAD are drawn meeting circles at P, C and Q, D respectively. If ∠PAB = 40°, ∠QAB = 30°, and tangents at P and Q intersect at T, find ∠PTQ.", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "C"},
    {"id": 93, "difficulty": "hard", "question": "In circle with center O, diameter AB. Tangent at B meets AC produced at T. If CT:TA = 2:3 and radius = 5 cm, find AT.", "options": {"A": "10 cm", "B": "12 cm", "C": "15 cm", "D": "18 cm"}, "correct_answer": "C"},
    {"id": 94, "difficulty": "hard", "question": "Two circles of radii 10 cm and 4 cm have their centers 20 cm apart. Find length of transverse common tangent.", "options": {"A": "8√3 cm", "B": "8√5 cm", "C": "8√6 cm", "D": "8√7 cm"}, "correct_answer": "C"},
    {"id": 95, "difficulty": "hard", "question": "From point P, tangents PA and PB to circle with center O. Another tangent touches circle at C and intersects PA at X, PB at Y. If perimeter of triangle PXY is 30 cm, find area of triangle OAB given OA = 6 cm.", "options": {"A": "18√3 cm²", "B": "24√3 cm²", "C": "36√3 cm²", "D": "48√3 cm²"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Two circles touch externally at P. Common external tangent touches circles at A and B. Another tangent from A to second circle touches at C. If AB = 8 cm, find area of triangle ABC.", "options": {"A": "8√2 cm²", "B": "16√2 cm²", "C": "24√2 cm²", "D": "32√2 cm²"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "Three circles with centers A, B, C touch each other externally. AB = 14 cm, BC = 15 cm, CA = 13 cm. Find radius of circle with center A.", "options": {"A": "6 cm", "B": "7 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "In circle with center O, chord AB subtends 90° at O. Tangents at A and B intersect at P. Line through P parallel to AB meets OA at C and OB at D. If radius = 10 cm, find area of quadrilateral ABDC.", "options": {"A": "100 cm²", "B": "150 cm²", "C": "200 cm²", "D": "250 cm²"}, "correct_answer": "C"},
    {"id": 99, "difficulty": "hard", "question": "Two circles intersect orthogonally. Sum of their radii is 17 cm and distance between centers is 13 cm. Find difference of radii.", "options": {"A": "5 cm", "B": "7 cm", "C": "9 cm", "D": "11 cm"}, "correct_answer": "B"},
    {"id": 100, "difficulty": "hard", "question": "From point P, tangents PA and PB to circle. Another circle with PA as diameter touches PB at Q. If PA = 8 cm, find area of triangle PQB.", "options": {"A": "8√3 cm²", "B": "16√3 cm²", "C": "24√3 cm²", "D": "32√3 cm²"}, "correct_answer": "B"}
  ]
}