Introduction to Three Dimensional Geometry — Important Questions
ICSE · Class 11 · Mathematics
59 important questions from Introduction to Three Dimensional Geometry for ICSE Class 11 Mathematics, with answers. Includes multiple choice questions.
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Important Questions from Introduction to Three Dimensional Geometry
Find the point on the x-axis that is equidistant from A(1, 2, 2) and B(7, 5, 8).
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(11/2, 0, 0)
Let the point be P(x, 0, 0). Then PA = PB. So √((x-1)^2 + (0-2)^2 + (0-2)^2) = √((x-7)^2 + (0-5)^2 + (0-8)^2). Squaring gives (x-1)^2 + 8 = (x-7)^2 + 89. Simplifying: x^2 - 2x + 9 = x^2 - 14x + 138, so 12x = 129 and x = 11/2. Therefore P = (11/2, 0, 0).
A point lies in the yz-plane and is equidistant from A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1). Find the point.
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(0, 1, 3)
Let P = (0, y, z). Equating PA and PB gives 3y - z = 0. Equating PB and PC gives 3y + z = 6. Solving these two equations gives y = 1 and z = 3. So the point is (0, 1, 3).
Find the coordinates of the centroid of triangle with vertices A(1, 2, 3), B(4, 5, 6) and C(7, 8, 9).
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(4, 5, 6)
The centroid is the average of the three vertices. So G = ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3) = (4, 5, 6).
Find the coordinates of the point that divides the line segment joining P(2, -3, 4) and Q(8, 0, 10) in the ratio 1:2 internally.
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(4, -2, 6)
Use the internal section formula with m:n = 1:2. Coordinates are ((1·8 + 2·2)/3, (1·0 + 2·(-3))/3, (1·10 + 2·4)/3) = (4, -2, 6).
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