{
  "title": "Distance Formula (Coordinate Geometry) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Distance between points (2,3) and (5,3) is ______. 📏", "options": {"A": "3", "B": "2", "C": "5", "D": "1"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Distance formula between (x1,y1) and (x2,y2) is √[(x2-x1)² + (y2-y1)²]. True?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Distance between (0,0) and (3,4) is ______.", "options": {"A": "5", "B": "7", "C": "1", "D": "25"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Points (1,2) and (1,5) have distance = ?", "options": {"A": "3", "B": "4", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Distance between (4,0) and (10,0) = ?", "options": {"A": "6", "B": "14", "C": "4", "D": "10"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "If both points have same y, distance = difference in ___.", "options": {"A": "x", "B": "y", "C": "origin", "D": "both"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "Distance from (2,2) to (5,6) uses formula √((5-2)² + (6-2)²) = √(9+16) = √25 = 5. True?", "options": {"A": "True", "B": "False", "C": "Maybe", "D": "Can’t say"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Points (-1, -1) and (2,2) distance?", "options": {"A": "√18", "B": "3", "C": "√8", "D": "4"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Distance between (0,0) and (0,7) = ?", "options": {"A": "7", "B": "0", "C": "1", "D": "49"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If distance = 0, the points are ______.", "options": {"A": "same", "B": "different", "C": "far", "D": "origin"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Distance between (a,b) and (c,d) formula?", "options": {"A": "√[(c-a)²+(d-b)²]", "B": "(c-a)+(d-b)", "C": "√(a²+b²)", "D": "c²+d²"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Points (3,4) and (3,9) distance?", "options": {"A": "5", "B": "6", "C": "13", "D": "0"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "If x1=x2, distance = |y2-y1|. True?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Distance between (-2,0) and (3,0) = ?", "options": {"A": "5", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Origin to (6,8) distance?", "options": {"A": "10", "B": "14", "C": "7", "D": "2"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Distance from (1,1) to (4,5) = ?", "options": {"A": "5", "B": "4", "C": "√13", "D": "3"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Points (0,5) and (0,-3) distance?", "options": {"A": "8", "B": "2", "C": "5", "D": "3"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Distance formula derived from ______ theorem.", "options": {"A": "Pythagoras", "B": "Thales", "C": "Euclid", "D": "Apollonius"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "If both coordinates same, distance = 0. Example?", "options": {"A": "(2,2) and (2,2)", "B": "(2,3) and (3,2)", "C": "(1,0) and (0,1)", "D": "(4,5) and (5,4)"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Distance between (7,1) and (7,1) = ?", "options": {"A": "0", "B": "1", "C": "7", "D": "√2"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find distance between ( -2, 3) and (4, -1).", "options": {"A": "√52", "B": "√40", "C": "6", "D": "8"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Points A(0,0), B(3,4), C(3,0) form right triangle. AB = ?", "options": {"A": "5", "B": "3", "C": "4", "D": "7"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Distance between (1/2, 3) and (4, 3) = ?", "options": {"A": "3.5", "B": "4", "C": "3", "D": "2.5"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If P(2,3), Q(5,7), then PQ = ?", "options": {"A": "5", "B": "4", "C": "√29", "D": "√13"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Points (a,0) and (0,a) distance?", "options": {"A": "a√2", "B": "a", "C": "2a", "D": "0"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Distance from (p,q) to origin?", "options": {"A": "√(p²+q²)", "B": "p+q", "C": "|p|+|q|", "D": "p²+q²"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Point A(3,k) and B(5,7) distance=√13. Find k.", "options": {"A": "5", "B": "6", "C": "8", "D": "9"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If distance between (4, y) and (9, 2) is 13, find y.", "options": {"A": "14 or -10", "B": "5", "C": "2", "D": "0"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Points (√2, √3) and (0,0) distance?", "options": {"A": "√5", "B": "√6", "C": "5", "D": "1"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Vertices (0,0), (6,0), (0,8) form triangle. Hypotenuse length?", "options": {"A": "10", "B": "14", "C": "12", "D": "8"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Distance between (-3, -4) and (2, 1) = ?", "options": {"A": "5√2", "B": "5", "C": "√50", "D": "10"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If A(1,2), B(4,6), C(7,2) is isosceles? AB=BC?", "options": {"A": "yes", "B": "no", "C": "equilateral", "D": "right"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Point equidistant from (2,3) and (6,9) is on ______.", "options": {"A": "perpendicular bisector", "B": "circle", "C": "line", "D": "origin"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Distance between (t, 2t) and (1, 2) is √5. Find t.", "options": {"A": "0 or 2", "B": "1", "C": "3", "D": "-1"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Length of line joining (-5, 1) and (3, 5) = ?", "options": {"A": "√80", "B": "8", "C": "10", "D": "4√5"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Points A(0,4), B(3,0). AB = ?", "options": {"A": "5", "B": "7", "C": "√7", "D": "1"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If P(k, 4), Q(2, 6) distance = 2√2, find k.", "options": {"A": "0 or 4", "B": "2", "C": "6", "D": "1"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Distance from (x, y) to (0,0) is 10. Then x²+y²=?", "options": {"A": "100", "B": "10", "C": "50", "D": "20"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Points (-2, -3) and (-5, -7) distance?", "options": {"A": "5", "B": "√13", "C": "4", "D": "√29"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If midpoint of (a, b) and (c, d) is origin, distance between them = 2√(a²+b²). True?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Depends"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Find k if distance between (k, 2) and (3, 4) is √8.", "options": {"A": "1 or 5", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "Points A(2,3), B(5,8), C(8,3) form which triangle?", "options": {"A": "isosceles", "B": "equilateral", "C": "scalene", "D": "right"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Distance of point (4,5) from origin = ?", "options": {"A": "√41", "B": "9", "C": "√89", "D": "3"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "Vertices (1,1), (4,4), (4,1) form ______ triangle.", "options": {"A": "right isosceles", "B": "equilateral", "C": "scalene", "D": "obtuse"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "Point on x-axis equidistant from (2,3) and (4,5) is ______.", "options": {"A": "(7,0)", "B": "(0,7)", "C": "(5,0)", "D": "(0,0)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "If (x, y) is equidistant from (a+b, b-a) and (a-b, a+b), then ______.", "options": {"A": "bx=ay", "B": "ax=by", "C": "x=y", "D": "a=b"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Distance between (sin θ, cos θ) and (cos θ, sin θ) = √2|sin θ - cos θ|. True?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "Points A(0,6), B(8,0), C(0,0) form triangle. Length of median from C to AB?", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Find distance between (a+b, a-b) and (a-b, a+b).", "options": {"A": "b√8", "B": "2b√2", "C": "a√2", "D": "0"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "hard", "question": "Point P(2,3), Q(10, y) distance=10. Find y.", "options": {"A": "9 or -3", "B": "5", "C": "7", "D": "11"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "Distance between (log a, log b) and (log b, log a) is?", "options": {"A": "√2 |log a - log b|", "B": "|log a - log b|", "C": "log(ab)", "D": "0"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "Vertices A(1, -2), B(3,6), C(5,10) are collinear if AB+BC=AC. Check.", "options": {"A": "collinear", "B": "not collinear", "C": "right triangle", "D": "isosceles"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Point (h, k) equidistant from (4, 3) and (2, 5) and also from (6, 1) and (0, 7). Find h+k.", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "Distance between (2 cos θ, 0) and (0, 2 sin θ) = ?", "options": {"A": "2", "B": "4", "C": "2√2", "D": "1"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Find k so that (k, 2-2k), (-k+1, 2k), (-4-k, 6-2k) are collinear.", "options": {"A": "k=1/2", "B": "k=2", "C": "k=0", "D": "k=1"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "Point P divides A(2,1) and B(5,8) in ratio 2:1. Distance AP?", "options": {"A": "√20", "B": "√13", "C": "5", "D": "10"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Distance between (at₁², 2at₁) and (at₂², 2at₂) = a|t₁-t₂|√[(t₁+t₂)²+4]. True?", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "If A(3,4), B(7,7), C(11,10) are points, AB:BC =?", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "From point (3,4), distance to x-axis = ?, to y-axis = ?", "options": {"A": "4, 3", "B": "3, 4", "C": "5, 5", "D": "0, 0"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "Points (1,2), (4, y), (x, 6) and (3,5) are vertices of parallelogram. Find x+y.", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "A"}
  ]
}