{
  "title": "Number_of_tangents_from_a_point(Circles) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "From point inside circle, number of tangents that can be drawn is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "From point on circle, number of tangents is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "From point outside circle, number of tangents is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If point is at distance equal to radius from center, it lies:", "options": {"A": "Inside circle", "B": "On circle", "C": "Outside circle", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Maximum number of tangents from point to circle is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "If distance from center < radius, point is:", "options": {"A": "Inside", "B": "On", "C": "Outside", "D": "At center"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "If distance from center > radius, point is:", "options": {"A": "Inside", "B": "On", "C": "Outside", "D": "At center"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "From center of circle, number of tangents is:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "D"},
    {"id": 9, "difficulty": "easy", "question": "Tangents from external point are:", "options": {"A": "Equal", "B": "Unequal", "C": "Parallel", "D": "Perpendicular"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If OP = 5 cm, radius = 3 cm (O is center, P is point), number of tangents from P:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "If OP = 4 cm, radius = 4 cm, number of tangents from P:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If OP = 3 cm, radius = 5 cm, number of tangents from P:", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Length of tangent from point 10 cm from center to circle of radius 6 cm is:", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "If two tangents from external point are 8 cm each and distance from center is 10 cm, radius is:", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "From point on circle, tangent is:", "options": {"A": "Radius", "B": "Diameter", "C": "Perpendicular to radius", "D": "Parallel to radius"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "If distance from center = radius, tangent makes angle with radius:", "options": {"A": "0°", "B": "45°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "For point inside circle, tangent:", "options": {"A": "Exists", "B": "Does not exist", "C": "May exist", "D": "Infinite exist"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "For point outside circle, two tangents are:", "options": {"A": "Parallel", "B": "Perpendicular", "C": "Equal", "D": "Unequal"}, "correct_answer": "C"},
    {"id": 19, "difficulty": "easy", "question": "The line joining external point to center bisects angle between:", "options": {"A": "Tangents", "B": "Radii", "C": "Both A and B", "D": "None"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "If PA and PB are tangents from P to circle with center O, then:", "options": {"A": "PA = PB", "B": "OA = OB", "C": "∠APO = ∠BPO", "D": "All of above"}, "correct_answer": "D"},
    {"id": 21, "difficulty": "medium", "question": "From point P, 25 cm from center O of circle radius 7 cm, find length of tangent.", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "If two tangents inclined at 60° are drawn from point P to circle radius 3 cm, find distance of P from center.", "options": {"A": "3 cm", "B": "6 cm", "C": "9 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "How many tangents can be drawn from point on major arc of circle to same circle?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If OP = 13 cm, radius = 5 cm, find perimeter quadrilateral OAPB.", "options": {"A": "34 cm", "B": "36 cm", "C": "38 cm", "D": "40 cm"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "If length of tangent from point to circle radius 6 cm is 8 cm, find distance from center.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "Two concentric circles have radii 5 cm and 3 cm. From point P on larger circle, tangent is drawn to smaller circle touching it at Q. Find PQ.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 27, "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": 28, "difficulty": "medium", "question": "How many tangents can be drawn from point on diameter (excluding endpoints)?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 29, "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": 30, "difficulty": "medium", "question": "If two tangents from external point are each 15 cm and distance between points of contact is 24 cm, find radius.", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If ∠OPA = 30°, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 32, "difficulty": "medium", "question": "How many common tangents can two concentric circles have?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 33, "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": "√132 cm"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "From point P, 17 cm from center O of circle radius 8 cm, find angle between tangents.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "If PA and PB are tangents from P to circle with center O, and ∠APB = 60°, find ∠OAB where A and B are points of contact.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "How many tangents can be drawn from point at distance equal to diameter from center?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "Two tangents TP and TQ are drawn from T to circle with center O. If ∠PTQ = 70°, find ∠OPQ.", "options": {"A": "20°", "B": "25°", "C": "30°", "D": "35°"}, "correct_answer": "D"},
    {"id": 38, "difficulty": "medium", "question": "From point on circle of radius 5 cm, tangent is drawn. Find length of tangent from point 13 cm from center to same circle.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 39, "difficulty": "medium", "question": "If two circles intersect at two points, number of common tangents is:", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If radius = 6 cm, PA = 8 cm, find OP.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 41, "difficulty": "medium", "question": "Two circles of radii 13 cm and 5 cm touch internally. Find length of common tangent.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "From point P, two tangents PA and PB are drawn. If PA = 12 cm, OP = 13 cm, find area ∆OAB where O is center.", "options": {"A": "30 cm²", "B": "60 cm²", "C": "90 cm²", "D": "120 cm²"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "How many tangents can be drawn from point at distance less than radius from center?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "Two circles of radii 5 cm and 3 cm have their centers 10 cm apart. Number of common tangents:", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "From point P outside circle, tangents PA and PB are drawn. Chord AB is drawn. If ∠PAB = 50°, find ∠APB.", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "medium", "question": "If two tangents from external point are inclined at 120°, find angle between radii to points of contact.", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Two circles touch externally. From point on common tangent at point of contact, tangents are drawn to circles. Prove they are equal.", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 48, "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": 49, "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": 50, "difficulty": "medium", "question": "From point P, 10 cm from center O of circle radius 6 cm, find area between chord of contact and tangents.", "options": {"A": "38.4 cm²", "B": "48 cm²", "C": "57.6 cm²", "D": "67.2 cm²"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "How many tangents can be drawn from origin to circle x²+y²+4x+4y+4=0?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "medium", "question": "Two tangents from point P to circle are perpendicular. If radius = 7 cm, find OP.", "options": {"A": "7√2 cm", "B": "14 cm", "C": "14√2 cm", "D": "21 cm"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "From point P on circle x²+y²=25, tangent is drawn. Find equation of tangent at (3,4).", "options": {"A": "3x+4y=25", "B": "4x+3y=25", "C": "3x-4y=25", "D": "4x-3y=25"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Two circles of radii 4 cm and 9 cm touch externally. Find length of transverse common tangent.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If ∠APB = 90°, and radius = 3 cm, find PA.", "options": {"A": "3 cm", "B": "3√2 cm", "C": "6 cm", "D": "6√2 cm"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "How many common tangents for circles x²+y²=9 and x²+y²-6x+8=0?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "medium", "question": "From point (2,3) to circle x²+y²=25, find length of tangent.", "options": {"A": "√12", "B": "√13", "C": "√14", "D": "√15"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two circles have radii 5 cm and 12 cm, distance between centers 25 cm. Number of common tangents:", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "From point P outside circle, tangents PA and PB are drawn. If ∠APB = 60°, and radius = 6 cm, find area quadrilateral OAPB.", "options": {"A": "36√3 cm²", "B": "48√3 cm²", "C": "60√3 cm²", "D": "72√3 cm²"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "How many tangents from (1,1) to circle x²+y²-2x-2y+1=0?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "Two circles touch internally. From point on larger circle, tangents to smaller circle are drawn. Maximum number of such tangents:", "options": {"A": "0", "B": "1", "C": "2", "D": "4"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If ∠APB = 80°, find reflex ∠AOB.", "options": {"A": "200°", "B": "220°", "C": "240°", "D": "260°"}, "correct_answer": "D"},
    {"id": 63, "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√7 cm", "D": "4√10 cm"}, "correct_answer": "C"},
    {"id": 64, "difficulty": "medium", "question": "From point P on director circle of x²+y²=16, number of tangents to original circle:", "options": {"A": "0", "B": "1", "C": "2", "D": "4"}, "correct_answer": "C"},
    {"id": 65, "difficulty": "medium", "question": "If two tangents from P to circle are at right angles and radius = 4 cm, find PA.", "options": {"A": "4 cm", "B": "4√2 cm", "C": "8 cm", "D": "8√2 cm"}, "correct_answer": "B"},
    {"id": 66, "difficulty": "medium", "question": "Two circles have no common tangents if one lies completely inside other without touching. Minimum distance between centers for radii 5 and 2:", "options": {"A": ">7", "B": "<3", "C": "3<d<7", "D": "d=0"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle. If ∠APB = 40°, find angle between chord AB and tangent at A.", "options": {"A": "20°", "B": "40°", "C": "60°", "D": "80°"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Two circles of radii 8 cm and 3 cm have centers 15 cm apart. Number of transverse common tangents:", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "C"},
    {"id": 69, "difficulty": "medium", "question": "From point P outside circle, length of tangent is 24 cm and distance from center is 26 cm. Find area enclosed between tangents and chord of contact.", "options": {"A": "240 cm²", "B": "280 cm²", "C": "320 cm²", "D": "360 cm²"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "How many tangents from (0,0) to circle x²+y²-6x-8y+25=0?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "Two circles intersect orthogonally if radii are 6 cm and 8 cm, distance between centers:", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle. Chord AB is drawn. If ∠PAB = 35°, find ∠APB.", "options": {"A": "55°", "B": "65°", "C": "70°", "D": "75°"}, "correct_answer": "C"},
    {"id": 73, "difficulty": "medium", "question": "Two circles of radii 5 cm and 12 cm touch externally. Find distance from point of contact to common tangent not through contact point.", "options": {"A": "60/13 cm", "B": "120/13 cm", "C": "180/13 cm", "D": "240/13 cm"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "From point on director circle of circle radius 5 cm, number of tangents:", "options": {"A": "0", "B": "1", "C": "2", "D": "4"}, "correct_answer": "C"},
    {"id": 75, "difficulty": "medium", "question": "If angle between tangents from P to circle is 60° and radius = 7 cm, find area ∆PAB where A,B are points of contact.", "options": {"A": "49√3/4 cm²", "B": "49√3/2 cm²", "C": "49√3 cm²", "D": "98√3 cm²"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "Two circles of radii 4 cm and 9 cm have centers d cm apart. For 2 direct common tangents, d satisfies:", "options": {"A": "d>13", "B": "d=13", "C": "5<d<13", "D": "d>5"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "From point P, tangents PA and PB are drawn to circle with center O. If ∠OAP = 30°, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "medium", "question": "Two circles touch internally. From point on larger circle, maximum tangents to smaller circle:", "options": {"A": "0", "B": "1", "C": "2", "D": "4"}, "correct_answer": "C"},
    {"id": 79, "difficulty": "medium", "question": "From point P, length of tangent to circle radius 5 cm is 12 cm. Find distance from P to farthest point on circle.", "options": {"A": "17 cm", "B": "18 cm", "C": "19 cm", "D": "20 cm"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "medium", "question": "Two circles have 3 common tangents. Their positions:", "options": {"A": "Intersect", "B": "Touch externally", "C": "Touch internally", "D": "One inside other"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "Prove that from point P outside circle, exactly two tangents can be drawn. What is length of tangent if OP=13 cm, r=5 cm?", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "From point P(h,k) to circle x²+y²=a², condition for tangents to be perpendicular:", "options": {"A": "h²+k²=2a²", "B": "h²+k²=3a²", "C": "h²+k²=4a²", "D": "h²+k²=5a²"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Two circles x²+y²-2x-4y+1=0 and x²+y²-8x-12y+43=0. Find number of common tangents.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "From point P, tangents PA and PB are drawn to circle with center O. Prove that quadrilateral OAPB is cyclic. Find its area if r=6 cm, OP=10 cm.", "options": {"A": "48 cm²", "B": "54 cm²", "C": "60 cm²", "D": "66 cm²"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Find locus of point from which tangents to circle x²+y²=a² are at right angles.", "options": {"A": "x²+y²=2a²", "B": "x²+y²=3a²", "C": "x²+y²=4a²", "D": "x²+y²=5a²"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Two circles of radii r1 and r2 have centers d apart. For 4 common tangents, condition on d:", "options": {"A": "d > r1+r2", "B": "d = r1+r2", "C": "|r1-r2| < d < r1+r2", "D": "d < |r1-r2|"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "From point P(2,3) to circle x²+y²-4x-6y+9=0, find angle between tangents.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 88, "difficulty": "hard", "question": "Prove that chord of contact from point P(x1,y1) to circle x²+y²=a² is xx1+yy1=a². Find its length if x1=3,y1=4,a=5.", "options": {"A": "8", "B": "9", "C": "10", "D": "12"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Two circles x²+y²+2gx+2fy+c=0 and x²+y²+2g'x+2f'y+c'=0 intersect orthogonally if:", "options": {"A": "2gg'+2ff'=c+c'", "B": "2gg'+2ff'=c-c'", "C": "2gg'+2ff'=c+c'+r1r2", "D": "2gg'+2ff'=c+c'-(r1²+r2²)"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "From point P, tangents PA and PB are drawn to circle with center O. If area ∆PAB = area quadrilateral OAPB, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "Find number of tangents from (1,2) to circle x²+y²-2x-4y+4=0.", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Two circles have equations x²+y²-4x-6y+9=0 and x²+y²+2x-8y+13=0. Find number of common tangents.", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "From point P on director circle of S=0, tangents are at right angles. Prove director circle equation is x²+y²=2a² for circle x²+y²=a².", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Find area of triangle formed by pair of tangents from (4,3) to circle x²+y²=9 and their chord of contact.", "options": {"A": "12", "B": "15", "C": "18", "D": "21"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Two circles of radii 5 and 12 have centers 13 apart. Prove they touch externally and find length of common tangent.", "options": {"A": "√120", "B": "√144", "C": "√156", "D": "√168"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "From point P(x1,y1) outside circle, equation of pair of tangents is SS1=T². For circle x²+y²=25 and point (7,1), find angle between tangents.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 97, "difficulty": "hard", "question": "Prove that for circles touching externally, there are 3 common tangents. For radii 4 and 9, find length of direct common tangent.", "options": {"A": "12", "B": "15", "C": "18", "D": "21"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Find locus of point from which tangents to circles x²+y²=4 and x²+y²-8x+12=0 are equal.", "options": {"A": "x=2", "B": "x=3", "C": "x=4", "D": "x=5"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "From point P, tangents PA and PB are drawn to circle x²+y²=a². If area ∆PAB is maximum, find ∠APB.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "Prove that from any point on director circle of given circle, two tangents can be drawn which are at right angles. What is director circle of x²+y²+2gx+2fy+c=0?", "options": {"A": "x²+y²+2gx+2fy+2c-g²-f²=0", "B": "x²+y²+2gx+2fy+3c-g²-f²=0", "C": "x²+y²+2gx+2fy+c-g²-f²=0", "D": "x²+y²+2gx+2fy=0"}, "correct_answer": "A"}
  ]
}