{
  "title": "Section Formula (Coordinate Geometry) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Find the coordinates of point dividing (2,3) and (4,5) in ratio 1:1 internally.", "options": {"A": "(3,4)", "B": "(4,3)", "C": "(5,6)", "D": "(6,5)"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "The point dividing (1,2) and (3,4) in ratio 2:1 internally is:", "options": {"A": "(7/3, 8/3)", "B": "(8/3, 7/3)", "C": "(5/3, 6/3)", "D": "(6/3, 5/3)"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Find coordinates of point dividing (4,5) and (7,8) in ratio 1:2 internally.", "options": {"A": "(5,6)", "B": "(6,5)", "C": "(7,8)", "D": "(8,7)"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "The midpoint formula is special case of section formula when ratio is:", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Find point dividing (0,0) and (6,9) in ratio 1:2 internally.", "options": {"A": "(2,3)", "B": "(3,2)", "C": "(4,6)", "D": "(6,4)"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "The coordinates of point dividing (x₁,y₁) and (x₂,y₂) in ratio m:n internally are:", "options": {"A": "((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))", "B": "((mx₁+nx₂)/(m+n), (my₁+ny₂)/(m+n))", "C": "((x₁+x₂)/2, (y₁+y₂)/2)", "D": "((x₁-x₂)/2, (y₁-y₂)/2)"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "Find point dividing (2,4) and (6,8) in ratio 3:1 internally.", "options": {"A": "(5,7)", "B": "(7,5)", "C": "(6,8)", "D": "(8,6)"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "The point dividing (1,1) and (4,4) in ratio 1:3 internally is:", "options": {"A": "(7/4, 7/4)", "B": "(5/2, 5/2)", "C": "(3/2, 3/2)", "D": "(2,2)"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Find coordinates of point dividing (3,6) and (9,12) in ratio 2:1 internally.", "options": {"A": "(7,10)", "B": "(10,7)", "C": "(8,11)", "D": "(11,8)"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "The point dividing (5,10) and (10,20) in ratio 1:4 internally is:", "options": {"A": "(6,12)", "B": "(12,6)", "C": "(8,16)", "D": "(16,8)"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Find point dividing (0,5) and (0,10) in ratio 2:3 internally.", "options": {"A": "(0,7)", "B": "(7,0)", "C": "(0,8)", "D": "(8,0)"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "The point dividing (8,0) and (0,8) in ratio 3:1 internally is:", "options": {"A": "(2,6)", "B": "(6,2)", "C": "(4,4)", "D": "(8,8)"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Find coordinates of point dividing (-1,-2) and (3,4) in ratio 1:1 internally.", "options": {"A": "(1,1)", "B": "(-1,-1)", "C": "(2,2)", "D": "(-2,-2)"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "The point dividing (2,2) and (8,8) in ratio 1:2 internally is:", "options": {"A": "(4,4)", "B": "(6,6)", "C": "(5,5)", "D": "(3,3)"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Find point dividing (10,20) and (20,40) in ratio 3:2 internally.", "options": {"A": "(14,28)", "B": "(28,14)", "C": "(16,32)", "D": "(32,16)"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "The coordinates for external division in ratio m:n are:", "options": {"A": "((mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n))", "B": "((mx₁-nx₂)/(m-n), (my₁-ny₂)/(m-n))", "C": "((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))", "D": "((x₁+x₂)/2, (y₁+y₂)/2)"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Find point dividing (1,2) and (3,4) in ratio 2:1 externally.", "options": {"A": "(5,6)", "B": "(6,5)", "C": "(4,5)", "D": "(5,4)"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "The point dividing (4,5) and (7,8) in ratio 1:2 externally is:", "options": {"A": "(1,2)", "B": "(2,1)", "C": "(3,4)", "D": "(4,3)"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Find coordinates of point dividing (0,0) and (6,9) in ratio 1:2 externally.", "options": {"A": "(-6,-9)", "B": "(-9,-6)", "C": "(-3,-4.5)", "D": "(-4.5,-3)"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "The point dividing (5,10) and (10,20) in ratio 3:2 externally is:", "options": {"A": "(20,40)", "B": "(40,20)", "C": "(15,30)", "D": "(30,15)"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find the coordinates of point dividing (2,3) and (5,7) in ratio 2:3 internally.", "options": {"A": "(16/5, 23/5)", "B": "(23/5, 16/5)", "C": "(19/5, 26/5)", "D": "(26/5, 19/5)"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Find point which divides line joining (1,2) and (3,4) externally in ratio 3:2.", "options": {"A": "(7,10)", "B": "(10,7)", "C": "(5,6)", "D": "(6,5)"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Find coordinates of point dividing (-2,-3) and (4,5) in ratio 3:4 internally.", "options": {"A": "(2/7, 3/7)", "B": "(3/7, 2/7)", "C": "(4/7, 5/7)", "D": "(5/7, 4/7)"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "In what ratio does point (4,5) divide segment joining (1,2) and (7,8)?", "options": {"A": "1:2", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Find ratio in which point (3,4) divides segment joining (1,2) and (5,6).", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "The point (11,14) divides line joining (2,3) and (5,6) in ratio:", "options": {"A": "3:1 externally", "B": "1:3 externally", "C": "3:1 internally", "D": "1:3 internally"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Find coordinates of point dividing (a,b) and (c,d) in ratio m:n internally.", "options": {"A": "((mc+na)/(m+n), (md+nb)/(m+n))", "B": "((ma+nc)/(m+n), (mb+nd)/(m+n))", "C": "((mc-na)/(m-n), (md-nb)/(m-n))", "D": "((ma-nc)/(m-n), (mb-nd)/(m-n))"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Find point dividing (3,5) and (8,10) in ratio 2:3 internally.", "options": {"A": "(5,7)", "B": "(7,5)", "C": "(6,8)", "D": "(8,6)"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "In what ratio does y-axis divide line joining (2,3) and (-4,5)?", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Find ratio in which point (0,3) divides line joining (-2,1) and (4,5).", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "The centroid of triangle divides median in ratio:", "options": {"A": "2:1", "B": "1:2", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Find coordinates of point dividing (2,4) and (6,8) in ratio 1:3 externally.", "options": {"A": "(0,2)", "B": "(2,0)", "C": "(4,6)", "D": "(6,4)"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "In what ratio does point (4,6) divide line joining (1,2) and (7,10)?", "options": {"A": "1:2", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Find point dividing (5,5) and (10,10) in ratio 3:2 externally.", "options": {"A": "(20,20)", "B": "(20,20)", "C": "(15,15)", "D": "(25,25)"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "The coordinates of centroid of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) are:", "options": {"A": "((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)", "B": "((x₁+x₂+x₃)/2, (y₁+y₂+y₃)/2)", "C": "((x₁+x₂)/2, (y₁+y₂)/2)", "D": "((x₂+x₃)/2, (y₂+y₃)/2)"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Find coordinates of point dividing (1,3) and (4,9) in ratio 2:1 internally.", "options": {"A": "(3,7)", "B": "(7,3)", "C": "(4,8)", "D": "(8,4)"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "In what ratio does point (2,0) divide line joining (-1,2) and (5,6)?", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Find point dividing (3,7) and (6,13) in ratio 1:2 externally.", "options": {"A": "(0,1)", "B": "(1,0)", "C": "(2,5)", "D": "(5,2)"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "The point (-1,5) divides line joining (2,3) and (5,6) in ratio:", "options": {"A": "1:2 externally", "B": "2:1 externally", "C": "1:2 internally", "D": "2:1 internally"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Find coordinates of centroid of triangle with vertices (1,2), (3,4), (5,6).", "options": {"A": "(3,4)", "B": "(4,3)", "C": "(5,6)", "D": "(6,5)"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Find coordinates of point dividing (a cosα, a sinα) and (a cosβ, a sinβ) in ratio m:n internally.", "options": {"A": "((a(m cosβ+n cosα))/(m+n), (a(m sinβ+n sinα))/(m+n))", "B": "((a(m cosα+n cosβ))/(m+n), (a(m sinα+n sinβ))/(m+n))", "C": "((a cosα cosβ), (a sinα sinβ))", "D": "((a), (a))"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "In what ratio does point (h,k) divide line joining (x₁,y₁) and (x₂,y₂)?", "options": {"A": "(h-x₁)/(x₂-h) = (k-y₁)/(y₂-k) = λ (say)", "B": "(x₂-h)/(h-x₁) = (y₂-k)/(k-y₁) = λ", "C": "(h-x₁)/(x₂-x₁) = (k-y₁)/(y₂-y₁) = λ", "D": "(x₂-h)/(x₂-x₁) = (y₂-k)/(y₂-y₁) = λ"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Find ratio in which line 3x+4y=7 divides line joining points (1,2) and (3,4).", "options": {"A": "3:5 externally", "B": "5:3 externally", "C": "3:5 internally", "D": "5:3 internally"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "Prove that points (1,2), (3,4), (5,6) are collinear using section formula.", "options": {"A": "(3,4) divides (1,2) and (5,6) in ratio 1:1", "B": "They lie on same line y=x+1", "C": "Slopes are equal", "D": "All of these"}, "correct_answer": "D"},
    {"id": 45, "difficulty": "hard", "question": "Find coordinates of point dividing (at₁², 2at₁) and (at₂², 2at₂) in ratio m:n internally.", "options": {"A": "((a(mt₂²+nt₁²))/(m+n), (2a(mt₂+nt₁))/(m+n))", "B": "((a(mt₁²+nt₂²))/(m+n), (2a(mt₁+nt₂))/(m+n))", "C": "((a(t₁²+t₂²))/2, (a(t₁+t₂)))", "D": "((a), (2a))"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "If point P divides AB internally in ratio 2:3 and Q divides AB externally in ratio 2:3, then:", "options": {"A": "P is between A and B, Q is outside", "B": "Both are between A and B", "C": "Both are outside", "D": "P is outside, Q is between"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Find ratio in which line joining (1,2) and (3,4) is divided by line x+y=5.", "options": {"A": "1:2 internally", "B": "2:1 internally", "C": "1:2 externally", "D": "2:1 externally"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "The coordinates of incenter of triangle using section formula involve:", "options": {"A": "Angle bisectors and side lengths", "B": "Medians", "C": "Altitudes", "D": "Perpendicular bisectors"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "If A(1,2), B(3,4), C(5,6) are vertices of triangle, find coordinates of centroid using section formula.", "options": {"A": "(3,4)", "B": "Centroid divides median in ratio 2:1", "C": "Both A and B", "D": "None of these"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "hard", "question": "Find point dividing (a secα, b tanα) and (a secβ, b tanβ) in ratio m:n internally.", "options": {"A": "((a(m secβ+n secα))/(m+n), (b(m tanβ+n tanα))/(m+n))", "B": "((a(m secα+n secβ))/(m+n), (b(m tanα+n tanβ))/(m+n))", "C": "((a secα secβ), (b tanα tanβ))", "D": "((a), (b))"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Find coordinates of point dividing (3,4) and (6,8) in ratio 2:3 internally.", "options": {"A": "(21/5, 32/5)", "B": "(32/5, 21/5)", "C": "(24/5, 36/5)", "D": "(36/5, 24/5)"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Find point dividing (2,5) and (5,11) externally in ratio 3:2.", "options": {"A": "(11,23)", "B": "(23,11)", "C": "(13,25)", "D": "(25,13)"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Find coordinates of point dividing (-3,-4) and (5,7) in ratio 4:3 internally.", "options": {"A": "(11/7, 16/7)", "B": "(16/7, 11/7)", "C": "(13/7, 18/7)", "D": "(18/7, 13/7)"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "In what ratio does point (5,6) divide segment joining (2,3) and (8,9)?", "options": {"A": "1:1", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Find ratio in which point (4,5) divides segment joining (1,2) and (7,8).", "options": {"A": "1:2", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "The point (13,16) divides line joining (3,4) and (5,6) in ratio:", "options": {"A": "5:1 externally", "B": "1:5 externally", "C": "5:1 internally", "D": "1:5 internally"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Find coordinates of point dividing (p,q) and (r,s) in ratio l:m internally.", "options": {"A": "((lr+mp)/(l+m), (ls+mq)/(l+m))", "B": "((lp+mr)/(l+m), (lq+ms)/(l+m))", "C": "((lr-mp)/(l-m), (ls-mq)/(l-m))", "D": "((lp-mr)/(l-m), (lq-ms)/(l-m))"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Find point dividing (4,6) and (9,11) in ratio 3:2 internally.", "options": {"A": "(7,9)", "B": "(9,7)", "C": "(8,10)", "D": "(10,8)"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "In what ratio does y-axis divide line joining (3,4) and (-6,8)?", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Find ratio in which point (0,4) divides line joining (-3,2) and (6,8).", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "hard", "question": "Find coordinates of point dividing (a cosθ, b sinθ) and (a cosφ, b sinφ) in ratio p:q internally.", "options": {"A": "((a(p cosφ+q cosθ))/(p+q), (b(p sinφ+q sinθ))/(p+q))", "B": "((a(p cosθ+q cosφ))/(p+q), (b(p sinθ+q sinφ))/(p+q))", "C": "((a cosθ cosφ), (b sinθ sinφ))", "D": "((a), (b))"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "hard", "question": "In what ratio does point (h,k) divide line joining (x₁,y₁) and (x₂,y₂) if h=(mx₂+nx₁)/(m+n)?", "options": {"A": "m:n", "B": "n:m", "C": "(h-x₁):(x₂-h)", "D": "(x₂-h):(h-x₁)"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "hard", "question": "Find ratio in which line 2x+3y=5 divides line joining points (1,1) and (4,7).", "options": {"A": "2:3 externally", "B": "3:2 externally", "C": "2:3 internally", "D": "3:2 internally"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "Prove that points (2,3), (4,5), (6,7) are collinear using section formula.", "options": {"A": "(4,5) divides (2,3) and (6,7) in ratio 1:1", "B": "They lie on line y=x+1", "C": "Area of triangle is zero", "D": "All of these"}, "correct_answer": "D"},
    {"id": 65, "difficulty": "hard", "question": "Find coordinates of point dividing (am², 2am) and (an², 2an) in ratio r:s internally.", "options": {"A": "((a(rm²+sn²))/(r+s), (2a(rm+sn))/(r+s))", "B": "((a(rm+sn))/(r+s), (2a(rm²+sn²))/(r+s))", "C": "((a(m²+n²))/2, (a(m+n)))", "D": "((a), (2a))"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "hard", "question": "If point R divides AB internally in ratio 3:4 and S divides AB externally in ratio 3:4, then:", "options": {"A": "R is between A and B, S is outside", "B": "Both are between A and B", "C": "Both are outside", "D": "R is outside, S is between"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "Find ratio in which line joining (2,3) and (5,6) is divided by line x-y+1=0.", "options": {"A": "1:2 internally", "B": "2:1 internally", "C": "1:2 externally", "D": "2:1 externally"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "The coordinates of centroid using section formula are derived from:", "options": {"A": "Intersection of medians", "B": "Intersection of altitudes", "C": "Intersection of angle bisectors", "D": "Intersection of perpendicular bisectors"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "hard", "question": "If A(2,4), B(4,6), C(6,8) are vertices of triangle, find coordinates of centroid using section formula.", "options": {"A": "(4,6)", "B": "Centroid divides median in ratio 2:1", "C": "Both A and B", "D": "None of these"}, "correct_answer": "C"},
    {"id": 70, "difficulty": "hard", "question": "Find point dividing (a cosecα, b cotα) and (a cosecβ, b cotβ) in ratio m:n internally.", "options": {"A": "((a(m cosecβ+n cosecα))/(m+n), (b(m cotβ+n cotα))/(m+n))", "B": "((a(m cosecα+n cosecβ))/(m+n), (b(m cotα+n cotβ))/(m+n))", "C": "((a cosecα cosecβ), (b cotα cotβ))", "D": "((a), (b))"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "Find coordinates of point dividing (5,7) and (8,13) in ratio 3:4 internally.", "options": {"A": "(44/7, 67/7)", "B": "(67/7, 44/7)", "C": "(47/7, 73/7)", "D": "(73/7, 47/7)"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Find point dividing (3,8) and (6,17) externally in ratio 4:3.", "options": {"A": "(15,44)", "B": "(44,15)", "C": "(18,47)", "D": "(47,18)"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "Find coordinates of point dividing (-4,-5) and (6,8) in ratio 5:2 internally.", "options": {"A": "(22/7, 30/7)", "B": "(30/7, 22/7)", "C": "(26/7, 36/7)", "D": "(36/7, 26/7)"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "In what ratio does point (6,7) divide segment joining (3,4) and (9,10)?", "options": {"A": "1:1", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "Find ratio in which point (5,6) divides segment joining (2,3) and (8,9).", "options": {"A": "1:1", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "The point (17,20) divides line joining (4,5) and (6,7) in ratio:", "options": {"A": "13:2 externally", "B": "2:13 externally", "C": "13:2 internally", "D": "2:13 internally"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Find coordinates of point dividing (u,v) and (w,x) in ratio c:d internally.", "options": {"A": "((cw+du)/(c+d), (cx+dv)/(c+d))", "B": "((cu+dw)/(c+d), (cv+dx)/(c+d))", "C": "((cw-du)/(c-d), (cx-dv)/(c-d))", "D": "((cu-dw)/(c-d), (cv-dx)/(c-d))"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "Find point dividing (6,9) and (11,16) in ratio 4:1 internally.", "options": {"A": "(10,13)", "B": "(13,10)", "C": "(11,15)", "D": "(15,11)"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "In what ratio does y-axis divide line joining (5,6) and (-10,12)?", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Find ratio in which point (0,5) divides line joining (-4,3) and (8,9).", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 81, "difficulty": "hard", "question": "Find coordinates of point dividing (a e^α, b e^{-α}) and (a e^β, b e^{-β}) in ratio p:q internally.", "options": {"A": "((a(p e^β+q e^α))/(p+q), (b(p e^{-β}+q e^{-α}))/(p+q))", "B": "((a(p e^α+q e^β))/(p+q), (b(p e^{-α}+q e^{-β}))/(p+q))", "C": "((a e^{α+β}), (b e^{-(α+β)}))", "D": "((a), (b))"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "In what ratio does point (h,k) divide line joining (x₁,y₁) and (x₂,y₂) if k=(my₂+ny₁)/(m+n)?", "options": {"A": "m:n", "B": "n:m", "C": "(k-y₁):(y₂-k)", "D": "(y₂-k):(k-y₁)"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Find ratio in which line 4x+5y=13 divides line joining points (2,1) and (7,6).", "options": {"A": "3:2 externally", "B": "2:3 externally", "C": "3:2 internally", "D": "2:3 internally"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Prove that points (3,4), (5,6), (7,8) are collinear using section formula.", "options": {"A": "(5,6) divides (3,4) and (7,8) in ratio 1:1", "B": "They lie on line y=x+1", "C": "Area is zero", "D": "All of these"}, "correct_answer": "D"},
    {"id": 85, "difficulty": "hard", "question": "Find coordinates of point dividing (ap², 2ap) and (aq², 2aq) in ratio r:s internally.", "options": {"A": "((a(rp²+sq²))/(r+s), (2a(rp+sq))/(r+s))", "B": "((a(rp+sq))/(r+s), (2a(rp²+sq²))/(r+s))", "C": "((a(p²+q²))/2, (a(p+q)))", "D": "((a), (2a))"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If point T divides AB internally in ratio 5:2 and U divides AB externally in ratio 5:2, then:", "options": {"A": "T is between A and B, U is outside", "B": "Both are between A and B", "C": "Both are outside", "D": "T is outside, U is between"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Find ratio in which line joining (3,4) and (6,7) is divided by line 2x-y=5.", "options": {"A": "1:1 internally", "B": "2:1 internally", "C": "1:1 externally", "D": "2:1 externally"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "The coordinates of circumcenter using section formula involve:", "options": {"A": "Perpendicular bisectors", "B": "Medians", "C": "Altitudes", "D": "Angle bisectors"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "If A(4,6), B(6,8), C(8,10) are vertices of triangle, find coordinates of centroid using section formula.", "options": {"A": "(6,8)", "B": "Centroid divides median in ratio 2:1", "C": "Both A and B", "D": "None of these"}, "correct_answer": "C"},
    {"id": 90, "difficulty": "hard", "question": "Find point dividing (a secα, b tanα) and (a secβ, b tanβ) in ratio m:n externally.", "options": {"A": "((a(m secβ-n secα))/(m-n), (b(m tanβ-n tanα))/(m-n))", "B": "((a(m secα-n secβ))/(m-n), (b(m tanα-n tanβ))/(m-n))", "C": "((a secα secβ), (b tanα tanβ))", "D": "((a), (b))"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "medium", "question": "Find coordinates of point dividing (7,10) and (10,16) in ratio 5:2 internally.", "options": {"A": "(64/7, 100/7)", "B": "(100/7, 64/7)", "C": "(70/7, 112/7)", "D": "(112/7, 70/7)"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "medium", "question": "Find point dividing (4,11) and (7,20) externally in ratio 5:3.", "options": {"A": "(13/2, 35/2)", "B": "(35/2, 13/2)", "C": "(15/2, 41/2)", "D": "(41/2, 15/2)"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "medium", "question": "Find coordinates of point dividing (-5,-7) and (7,11) in ratio 6:5 internally.", "options": {"A": "(17/11, 19/11)", "B": "(19/11, 17/11)", "C": "(23/11, 29/11)", "D": "(29/11, 23/11)"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "medium", "question": "In what ratio does point (7,8) divide segment joining (4,5) and (10,11)?", "options": {"A": "1:1", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "medium", "question": "Find ratio in which point (6,7) divides segment joining (3,4) and (9,10).", "options": {"A": "1:1", "B": "2:1", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "medium", "question": "The point (19,22) divides line joining (5,6) and (7,8) in ratio:", "options": {"A": "7:1 externally", "B": "1:7 externally", "C": "7:1 internally", "D": "1:7 internally"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "medium", "question": "Find coordinates of point dividing (f,g) and (h,i) in ratio j:k internally.", "options": {"A": "((jh+kf)/(j+k), (ji+kg)/(j+k))", "B": "((jf+kh)/(j+k), (jg+ki)/(j+k))", "C": "((jh-kf)/(j-k), (ji-kg)/(j-k))", "D": "((jf-kh)/(j-k), (jg-ki)/(j-k))"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "medium", "question": "Find point dividing (8,12) and (13,19) in ratio 5:3 internally.", "options": {"A": "(89/8, 137/8)", "B": "(137/8, 89/8)", "C": "(95/8, 147/8)", "D": "(147/8, 95/8)"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "medium", "question": "In what ratio does y-axis divide line joining (7,8) and (-14,16)?", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "medium", "question": "Find ratio in which point (0,6) divides line joining (-5,4) and (10,10).", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"}
  ]
}