Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

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Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers

Question 1.
Prove by using distance formula that the A(1, 2, 3), B(-1, -1, -1) and C(3, 5, 7) are collinear.
Answer:
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 1
Now BC = AB + AC
Thus A, B, C are collinear.

Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

Question 2.
Verify the following: (3 score each)

  1. (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an isosceles triangle.
  2. (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right angled triangle.
  3. (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parllelogram.

Answer:
1. Let A(0, 7, -10), B(1, 6, -6) and C(4, 9, -6) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 2
Now AB = BC, thus ABC is an isosceles triangle.

2. Let A(0, 7, 10), B(-1, 6, 6) and C(-4, 9, 6) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 3
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 4
Now AC2 = AB2 + BC2, thus ABC is a right triangle.

3. Let A(-1, 2, 1), B(1, -2, 5), C(4, -7, 8) and D(2, -3, 4) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 5
Now AB = CD, BC = AD, AC ≠ BD, thus A, B, C, D are vertices of a parallelogram.

Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

Question 3.
Find the equation of set points which are equidistant from the points (1, 2, 3) and (3, 2, -1).
Answer:
Let P(x,y,z) be any point which is equidistant from the point A(1, 2, 3) and B (3, 2, -1).
Given; PA = PB
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 6
(x – 1)2 + (y – 2)2 + (z – 3)2
= (x – 3)2 + (y – 2)2 + (z + 1)2
= x2 – 2x + 1 + y2 – 4y + 4 + z2 – 6z + 9
= x2 – 6x + 9 + y2 – 4y + 4 + z2 + 2z + 1 – 2x + 14 – 6z = -6x + 14 + 2z
⇒ 4x – 8z = 0
⇒ x – 2z = 0.

Question 4.
Find the coordinate of the point which divides the line segment joining the points (3, -2, 5) and (3, 4, 2) in the ratio 2:1 (3 score each)

  1. 2:1 internally
  2. 2:1 externally

Answer:
1. Let P(x, y, z) be any point which divides the line segment joining points A(3, -2, 5) and B (3, 4, 2) in the ratio 2:1 internally.
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 7
Therefore coordinates of P are (3, 2, 3).

2. Let P(x, y, z) be any point which divides the line segment joining points A(3, -2, 5) and B (3, 4, 2) in the ratio 2:1 externally.
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 8
Therefore coordinates of P are (3, 10, -1).

Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

Question 5.
Find the ratio in which the line joining the points (1, 2, 3) and (-3, 4, -5) is divided by the xy-plane.
Answer:
Let the line joining the points A(1, 2, 3) and B(-3, 4, -5) is divided by the xy-plane in the ratio k:1.
Then the coordinate
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 9
Since the point lies on xy-axis, we have;
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 10
Thus the required ratio is \(\frac{3}{5}\); ie, 3:5.

Question 6.
Find the coordinates of the points which trisect the line segment joining the points P(4, 2, -6) and Q (10, -16, 6).
Answer:
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 11
Let R and S be two points which trisect the line join of PQ. Therefore PR = RS = SQ Then coordinate of R is
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 12
= (6, -4, -2)
Then coordinate of S is
Plus One Maths Introduction to Three Dimensional Geometry Three Mark Questions and Answers 13
= (8, -10, 2).

Plus One Maths Introduction to Three Dimensional Geometry Practice Problems Questions and Answers

Question 1.
Find the distance between the following pair of points: (1 score each)

  1. (2, 3, 5) and (4, 3, 1)
  2. (-3, 7, 2) and (2, 4, -1)
  3. (-1, 3, -4) and (1, -3, 4)

Answer:
1. Let A(2, 3, 5) and B(4, 3, 1) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Practice Problems Questions and Answers 14

2. Let A(-3, 7, 2) and B(2, 4, -1) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Practice Problems Questions and Answers 15

Plus One Maths Chapter Wise Questions and Answers Chapter 12 Introduction to Three Dimensional Geometry

3. Let A(-1, 3, -4) and B(1, -3, 4) be the two points.
Plus One Maths Introduction to Three Dimensional Geometry Practice Problems Questions and Answers 16

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