CBSE 11th Standard Maths Subject Introduction to Three Dimensional Geometry Ncert Exemplar 2 Marks Questions 2021
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CBSE 11th Standard Maths Subject Introduction to Three Dimensional Geometry Ncert Exemplar 2 Marks Questions 2021
11th Standard CBSE
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Reg.No. :
Mathematics
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Let L,M,N be the feet of the perpendiculars drawn from a point P(3,4,5) on the X,Y and Z-axes respectively.Find the coordinates of L, M and N.
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Find the centroid of a triangle, the mid-point of whose sides are D(1, 2, -3), E (3, 0, 1) and F(-1, 1, -4).
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Find the point on X-axis which is equidistant from the points A(3,2,2) and B(5,5,4)
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Find the coordinates of a point equidistant from the four points O(0,0,0), A(l,0,0), B(0,m,0), and C(0,0,n)
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Let L,M,N be the feet of the perpendicules drawn from the point P (3,4,5) on the XY,YZ and ZX-planes, respectively. Find the distance of these points L,M,N from the point P.
2 Marks
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CBSE 11th Standard Maths Subject Introduction to Three Dimensional Geometry Ncert Exemplar 2 Marks Questions 2021 Answer Keys
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L(3,0,0), M(0,4,0) and N(0,0,5)
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The centroid of a triangle is equal to the centroid of the triangle formed by mid-points of its sides.
(1, 1, -2)
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Let the point on X-axis be P(x,0,0).
\(Then,\quad { (x-3) }^{ 2 }+{ (0-2) }^{ 2 }+{ (0-2) }^{ 2 }\)
\(={ (x-5) }^{ 2 }+{ (0-5) }^{ 2 }+{ (0-4) }^{ 2 }\)
\(Ans.\left( \frac { 49 }{ 4 } ,0,0 \right) \) -
\(Let\quad P(x,y,z)\quad be\quad required\quad point.\)
\(Then,\quad OP=PA=PB=PC\)
\(Now,\quad OP=PA\Rightarrow { OP }^{ 2 }={ PA }^{ 2 }\)
\(\Rightarrow { (0-x) }^{ 2 }+{ (0-y) }^{ 2 }+{ (0-z) }^{ 2 }={ x-1 }^{ 2 }+{ y-0 }^{ 2 }+{ (z-0) }^{ 2 }\)
\( \Rightarrow { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }={ x }^{ 2 }-2lx+{ l }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }\)
\(\Rightarrow 2lx={ l }^{ 2 }\Rightarrow x=\frac { 1 }{ 2 } \)
\(Similarly,\quad OP=PB\Rightarrow y=\frac { m }{ 2 } and\quad OP=PC\Rightarrow x=\frac { n }{ 2 } \)
\(Hence,\quad the\quad coordinates\quad of\quad the\quad required\quad point\quad are\quad \left( \frac { 1 }{ 2 } ,\frac { m }{ 2 } ,\frac { n }{ 2 } \right) .\) -
L is the foot of perpendicular drwn from the point P(3,4,5) to the XY-plane.
Therefore, the coordinate of the point L are (3,4,0). The distance between the points (3,4,5) and (3,4,0) is 5 units. Similarly, the lengths of the foot of perpendiculars on YZ and ZX-planes are 3 and 4 units, respectively.
2 Marks
