{
  "title": "Properties of Tangents - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "A tangent to a circle makes what angle with the radius at the point of contact?", "options": {"A": "60°", "B": "45°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "How many tangents can be drawn to a circle from a point outside the circle?", "options": {"A": "1", "B": "2", "C": "3", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "If two tangents are drawn from an external point to a circle, they are:", "options": {"A": "Parallel", "B": "Equal in length", "C": "Perpendicular", "D": "Unequal"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "The length of tangent from a point P to a circle with center O is 5 cm. If OP = 13 cm, what is the radius?", "options": {"A": "12 cm", "B": "8 cm", "C": "10 cm", "D": "15 cm"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "A line that touches a circle at exactly one point is called a:", "options": {"A": "Secant", "B": "Chord", "C": "Tangent", "D": "Diameter"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "If two circles touch internally, how many common tangents do they have?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "The point where tangent touches the circle is called:", "options": {"A": "Center", "B": "Point of contact", "C": "Midpoint", "D": "Vertex"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "If a line is perpendicular to the radius at its outer endpoint, then the line is:", "options": {"A": "A secant", "B": "A tangent", "C": "A chord", "D": "A diameter"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "From a point Q, the length of tangent to a circle is 12 cm and distance from center is 13 cm. Radius is:", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If two circles touch externally, how many common tangents do they have?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "In the given figure, PT is tangent to circle with center O. If ∠TPO = 30°, what is ∠POT?", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "A tangent at any point of a circle is perpendicular to the radius through the:", "options": {"A": "Center", "B": "Point of contact", "C": "Midpoint", "D": "Any point"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "If PA and PB are tangents from point P to a circle with center O, then triangle OAP is:", "options": {"A": "Isosceles", "B": "Right angled", "C": "Equilateral", "D": "Scalene"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "The number of tangents that can be drawn to a circle from a point inside it is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "In a circle with center O, PA and PB are tangents from point P. If ∠APB = 60°, what is ∠AOB?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "180°"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "The maximum number of tangents that can be drawn to a circle is:", "options": {"A": "1", "B": "2", "C": "Infinite", "D": "0"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "If two circles are concentric, how many common tangents do they have?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "A line intersecting a circle in two points is called:", "options": {"A": "Tangent", "B": "Secant", "C": "Diameter", "D": "Radius"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "In triangle OAP where OA is radius and AP is tangent, which angle is 90°?", "options": {"A": "∠OAP", "B": "∠OPA", "C": "∠AOP", "D": "∠OAP and ∠OPA"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "If two tangents from an external point are 8 cm each, and they make 60° angle, distance between points of contact is:", "options": {"A": "4 cm", "B": "8 cm", "C": "12 cm", "D": "16 cm"}, "correct_answer": "B"},
    
    {"id": 21, "difficulty": "medium", "question": "From a point P outside a circle with center O, tangents PA and PB are drawn. If OP = 10 cm and radius = 6 cm, find PA.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 22, "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": "6 cm", "B": "7 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "In a circle with center O, PA and PB are tangents. If ∠APB = 50°, find ∠OAB.", "options": {"A": "25°", "B": "40°", "C": "50°", "D": "65°"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "PQ is a tangent to a circle at Q. PR is a secant intersecting circle at R and S. If PQ = 6 cm, PR = 4 cm, find RS.", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "D"},
    {"id": 25, "difficulty": "medium", "question": "Two circles of radii 8 cm and 3 cm touch externally. Find length of direct common tangent.", "options": {"A": "√60 cm", "B": "√96 cm", "C": "√120 cm", "D": "√140 cm"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "From an external point P, tangents PA and PB are drawn to a circle with center O. If ∠APB = 70°, find ∠AOB.", "options": {"A": "70°", "B": "110°", "C": "140°", "D": "160°"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "AB and AC are tangents to a circle from point A. If AB = 7 cm and ∠BAC = 60°, find radius if distance from A to center is 14 cm.", "options": {"A": "7√3 cm", "B": "7 cm", "C": "14 cm", "D": "14√3 cm"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Two circles of radii 5 cm and 12 cm intersect. Their common chord is 8 cm. Find distance between centers.", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "19 cm"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "In a circle with center O, PQ is a tangent at Q. If OQ = 5 cm and OP = 13 cm, find PQ.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "Two tangents TP and TQ are drawn from T to a circle with center O. If ∠PTQ = 60° and radius = 7 cm, find OT.", "options": {"A": "7 cm", "B": "14 cm", "C": "7√3 cm", "D": "14√3 cm"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "A point P is 25 cm from center O of circle of radius 7 cm. Find length of tangent from P to circle.", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 32, "difficulty": "medium", "question": "Two circles touch externally. Sum of their radii is 14 cm and distance between centers is 10 cm. Find radii.", "options": {"A": "6 cm, 8 cm", "B": "4 cm, 10 cm", "C": "3 cm, 11 cm", "D": "5 cm, 9 cm"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "In a circle with center O, AB is diameter. From point P on tangent at B, PA is drawn meeting circle at Q. If ∠PAB = 30°, find ∠AQB.", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "Two circles of radii 13 cm and 5 cm have direct common tangent of length 24 cm. Find distance between centers.", "options": {"A": "26 cm", "B": "28 cm", "C": "30 cm", "D": "32 cm"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "From point P outside circle, two tangents PA and PB are drawn. If PA = 8 cm and ∠APB = 60°, find radius if OP = 10 cm.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "In a circle with center O, chord AB = 24 cm. Distance from O to AB is 5 cm. Find length of tangent from A if OA = 13 cm.", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Two circles intersect at P and Q. Their centers are O and O'. If ∠OPO' = 70°, find ∠OQO'.", "options": {"A": "70°", "B": "110°", "C": "140°", "D": "160°"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR is drawn parallel to tangent PQ. If ∠PTR = 40°, find ∠RQT.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "Two circles with radii 5 cm and 3 cm have their centers 10 cm apart. Find length of transverse common tangent.", "options": {"A": "6 cm", "B": "8 cm", "C": "√84 cm", "D": "√96 cm"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "In a circle with center O, PA and PB are tangents. If ∠APB = 80°, find ∠OAP.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "AB is diameter of circle. Tangent at point P meets AB produced at T. If ∠BPT = 30°, find ∠PBA.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "Two concentric circles have radii 13 cm and 5 cm. Find length of chord of larger circle which is tangent to smaller circle.", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "medium", "question": "From point A outside circle, AB and AC are tangents. If BC = 8 cm and radius = 3 cm, find AB.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "In a circle, two parallel chords are on same side of center. Their lengths are 24 cm and 10 cm, radius is 13 cm. Find distance between chords.", "options": {"A": "5 cm", "B": "7 cm", "C": "9 cm", "D": "11 cm"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TA makes angle 50° with tangent. Find angle subtended by chord TA at center.", "options": {"A": "50°", "B": "80°", "C": "100°", "D": "130°"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "Two circles touch internally. Distance between centers is 4 cm. If radius of larger circle is 7 cm, find radius of smaller circle.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "In a circle with center O, PA is tangent and PB is secant. If PA = 6 cm and PB = 9 cm, find length of chord BC if P-B-C are collinear.", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "Two circles intersect at A and B. Their radii are 5 cm and 3 cm. If AB = 4 cm, find distance between centers.", "options": {"A": "2√5 cm", "B": "3√3 cm", "C": "4√2 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "From point P outside circle, tangent PT and secant PAB are drawn. If PT = 8 cm and PA = 4 cm, find AB.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "In a circle, chord AB = 16 cm. Distance from center to AB is 6 cm. Find length of tangent from A if distance from A to center is 10 cm.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "medium", "question": "Two circles with radii R and r touch externally. Distance between centers is 15 cm. If R = 9 cm, find r.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "medium", "question": "PA and PB are tangents to circle from point P. If ∠APB = 70° and radius = 5 cm, find length of tangent.", "options": {"A": "5 tan 35°", "B": "5 cot 35°", "C": "5 sin 35°", "D": "5 cos 35°"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "In a circle with center O, PQ is diameter. Tangent at Q meets PR at T where R is on circle. If ∠QPR = 40°, find ∠QTR.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "medium", "question": "Two circles of radii 8 cm and 3 cm have their centers 13 cm apart. Find length of direct common tangent.", "options": {"A": "12 cm", "B": "√120 cm", "C": "√133 cm", "D": "√144 cm"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "From point A, two tangents AB and AC are drawn to circle with center O. If ∠BAC = 50°, find ∠BOC.", "options": {"A": "100°", "B": "115°", "C": "130°", "D": "140°"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "Chord AB of circle with center O is 12 cm. Perpendicular from O to AB is 5 cm. Find length of tangent from B if OB = 13 cm.", "options": {"A": "5 cm", "B": "12 cm", "C": "13 cm", "D": "17 cm"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "medium", "question": "Two circles touch internally. Radius of larger circle is 10 cm, distance between centers is 6 cm. Find radius of smaller circle.", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR makes 35° with tangent. Find ∠TOR where O is center.", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "Two circles intersect at A and B. Their radii are 10 cm and 8 cm. If distance between centers is 12 cm, find AB.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "medium", "question": "From point P outside circle, PA is tangent, PBC is secant. If PA = 6 cm, PB = 2 cm, find PC.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "18 cm"}, "correct_answer": "D"},
    {"id": 61, "difficulty": "medium", "question": "In a circle, chord AB = 30 cm. Distance from center to AB is 8 cm. Find length of tangent from A if OA = 17 cm.", "options": {"A": "8 cm", "B": "15 cm", "C": "17 cm", "D": "20 cm"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Two circles of radii 7 cm and 1 cm touch externally. Find length of direct common tangent.", "options": {"A": "4√3 cm", "B": "4√5 cm", "C": "4√6 cm", "D": "4√7 cm"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "PA and PB are tangents from point P to circle with center O. If ∠OAB = 25°, find ∠APB.", "options": {"A": "25°", "B": "50°", "C": "65°", "D": "70°"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "Chord AB of circle with center O is 16 cm. OM ⟂ AB where M is midpoint. If OM = 6 cm, find length of tangent from A if OA = 10 cm.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 65, "difficulty": "medium", "question": "Two circles touch internally. Sum of radii is 18 cm, difference is 8 cm. Find distance between centers.", "options": {"A": "5 cm", "B": "8 cm", "C": "10 cm", "D": "13 cm"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR when produced meets tangent at P. If ∠PTR = 30° and ∠RPT = 50°, find ∠TRP.", "options": {"A": "50°", "B": "60°", "C": "80°", "D": "100°"}, "correct_answer": "D"},
    {"id": 67, "difficulty": "medium", "question": "Two circles of radii 5 cm and 12 cm have transverse common tangent of length 15 cm. Find distance between centers.", "options": {"A": "13 cm", "B": "17 cm", "C": "20 cm", "D": "25 cm"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle. If PA = 8 cm and ∠APB = 60°, find radius if OP = 10 cm.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "medium", "question": "In a circle, chord AB = 24 cm. Distance from center to AB is 5 cm. Find length of tangent from B if distance from B to center is 13 cm.", "options": {"A": "5 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "Two circles touch externally. Their radii are in ratio 3:2. Distance between centers is 15 cm. Find radii.", "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": 71, "difficulty": "medium", "question": "PT is tangent to circle at T. Chord TR subtends 80° at center. Find ∠PTR.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Two concentric circles have radii 17 cm and 8 cm. Find length of chord of larger circle which touches smaller circle.", "options": {"A": "30 cm", "B": "32 cm", "C": "34 cm", "D": "36 cm"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "From point A, tangents AB and AC are drawn to circle. If BC = 10 cm and radius = 5 cm, find AB.", "options": {"A": "5√3 cm", "B": "10 cm", "C": "10√2 cm", "D": "12 cm"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "Two circles intersect at A and B. Their radii are 13 cm and 15 cm. Distance between centers is 14 cm. Find AB.", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "PA is tangent to circle at A. Chord AB makes 40° with tangent. Find angle subtended by AB at center.", "options": {"A": "40°", "B": "80°", "C": "100°", "D": "140°"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "Two circles touch internally. Radius of larger circle is 12 cm, distance between centers is 8 cm. Find radius of smaller circle.", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 77, "difficulty": "medium", "question": "From point P, tangent PT and secant PAB are drawn. If PT = 12 cm and PA = 8 cm, find AB.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "medium", "question": "Two circles of radii 10 cm and 6 cm have their centers 20 cm apart. Find length of direct common tangent.", "options": {"A": "12 cm", "B": "16 cm", "C": "√336 cm", "D": "√384 cm"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "medium", "question": "In a circle with center O, PA is tangent. Chord AB when produced meets tangent at P. If ∠PAB = 30°, find ∠AOB.", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "medium", "question": "Two circles touch externally. Sum of areas is 130π cm². Distance between centers is 14 cm. Find radii if they 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": 81, "difficulty": "hard", "question": "Two circles with centers O and O' intersect at A and B. If ∠OAO' = 40° and ∠OBO' = 70°, find angle between tangents at A to the two circles.", "options": {"A": "30°", "B": "40°", "C": "50°", "D": "60°"}, "correct_answer": "D"},
    {"id": 82, "difficulty": "hard", "question": "From an external point P, tangents PA and PB are drawn to a circle with center O. If OP is diameter of another circle passing through A and B, and radius of first circle is 6 cm, find OP.", "options": {"A": "6√2 cm", "B": "12 cm", "C": "12√2 cm", "D": "18 cm"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "Two circles of radii 5 cm and 12 cm intersect orthogonally. Find distance between their centers.", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "19 cm"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "In a circle with center O, PA and PB are tangents. A line through P parallel to AB meets OA and OB at C and D respectively. If radius = 5 cm and OP = 13 cm, find CD.", "options": {"A": "60/13 cm", "B": "120/13 cm", "C": "130/13 cm", "D": "150/13 cm"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "Three circles with radii 3 cm, 4 cm, and 5 cm touch each other externally. Find area of triangle formed by joining their centers.", "options": {"A": "12 cm²", "B": "18 cm²", "C": "24 cm²", "D": "36 cm²"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "Two circles intersect at A and B. Through A, line PAQ is drawn meeting circles at P and Q. If tangents at P and Q intersect at T, prove that P, B, Q, T are concyclic. Find ∠PTQ if ∠APB = 50° and ∠AQB = 60°. (Assume the configuration)", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "C"},
    {"id": 87, "difficulty": "hard", "question": "In a circle with center O, AB is diameter. From point T on tangent at B, TA is drawn meeting circle at C. If ∠BTA = 20°, find ∠BOC.", "options": {"A": "40°", "B": "60°", "C": "80°", "D": "100°"}, "correct_answer": "D"},
    {"id": 88, "difficulty": "hard", "question": "Two circles of radii 8 cm and 3 cm have their centers 13 cm apart. Find length of common chord when they intersect.", "options": {"A": "48/13 cm", "B": "96/13 cm", "C": "120/13 cm", "D": "144/13 cm"}, "correct_answer": "B"},
    {"id": 89, "difficulty": "hard", "question": "From point P outside circle, tangents PA and PB are drawn. Through midpoint M of AB, line PM is drawn meeting circle at C and D (C closer to P). If PM = 9 cm and MC = 4 cm, find radius.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "Two circles touch internally at A. Chord BC of larger circle touches smaller circle at D. If ∠BAC = 60°, find ratio BD:DC.", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "In a circle with center O, PA and PB are tangents. A line through P parallel to AB meets OA at C and OB at D. If PA = 12 cm and radius = 5 cm, find area of trapezium ABDC.", "options": {"A": "60 cm²", "B": "120 cm²", "C": "180 cm²", "D": "240 cm²"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Two circles intersect orthogonally. If radii are 6 cm and 8 cm, find length of common chord.", "options": {"A": "48/5 cm", "B": "96/5 cm", "C": "24/5 cm", "D": "12/5 cm"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "From point P, tangents PA and PB are drawn to circle with center O. A third tangent touches circle at C and intersects PA and PB at Q and R. If perimeter of triangle PQR is 24 cm, find PA.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "D"},
    {"id": 94, "difficulty": "hard", "question": "Two circles of radii 5 cm and 12 cm have direct common tangent of length 15 cm. Find distance between centers if they are on same side of tangent.", "options": {"A": "√106 cm", "B": "√154 cm", "C": "√194 cm", "D": "√234 cm"}, "correct_answer": "C"},
    {"id": 95, "difficulty": "hard", "question": "In a circle with center O, AB is chord. Tangents at A and B intersect at P. If ∠APB = 50°, find ∠AOB.", "options": {"A": "115°", "B": "125°", "C": "130°", "D": "140°"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Two circles touch externally at P. Common tangent touches circles at A and B. Another tangent from A to second circle touches it at C. If AB = 6 cm, find AC.", "options": {"A": "3 cm", "B": "3√2 cm", "C": "4 cm", "D": "4√2 cm"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "Three equal circles touch each other externally. Find ratio of area of triangle formed by centers to area of sector of one circle between two common tangents.", "options": {"A": "√3:π", "B": "2√3:π", "C": "3√3:π", "D": "4√3:π"}, "correct_answer": "A"},
    {"id": 98, "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 = 8 cm and radius = 5 cm, find BC.", "options": {"A": "6.4 cm", "B": "8 cm", "C": "9.6 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 99, "difficulty": "hard", "question": "Two circles of radii 7 cm and 1 cm have direct common tangent AB (A on larger circle). Find distance from A to point where line joining centers meets tangent if centers are 10 cm apart.", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "B"},
    {"id": 100, "difficulty": "hard", "question": "In a circle with center O, PA and PB are tangents. A line through P parallel to AB meets OA at C. If OA = 5 cm and OP = 13 cm, find ratio of areas of triangles OAP and OCP.", "options": {"A": "169:144", "B": "144:25", "C": "169:25", "D": "25:144"}, "correct_answer": "A"}
  ]
}