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Math Assignment Class XII Ch-11 | 3-Dimensional Geometry
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Math. Assignment Class XII Chapter 11
Question 2:
Question 3:
Hint: Two parallel lines have same direction ratio.
Answer:
Question 4: Find angle between pair of lines
and
Answer: 90o
Question 5:
Answer: and the point is either
Answer:
Question 7:
and
Answer:
Question 8: Show that the following lines are coplanar.
where t = 3λ
- Take any arbitrary point on the line AB in terms of λ and let it is the coordinate of point L
- Find the direction ratios of line PL
- Find direction ratio of line AB
- Line AB 丄 PQ so use perpendicularity condition to find the value of λ
- Putting the value of λ to find the coordinates of Point L.
- Point L is the foot of perpendicular in line AB.
- Let us take point Q the image of point P in the line AB.
- So point L is the mid point of PQ, and by using mid point formula find the coordinates of point Q
Answer:
intersect. Also find their point of intersection.
- Find arbitrary points on both the lines
- Compairing x, y and z-coordinates we get three equations.
- Solve the first two equations to find the values of λ and μ
- Putting the values of λ and μ in equation three.
- If we get LHS = RHS the the given lines intersect each other.
- Putting the value of λ or μ in arbitrary points we get the point of intersection of two lines.
Answer: Point of intersection (-1, - 6, -12)
Solution Algorithm
- Find
- If then lines are coplanar and so lines are intersecting lines.
- Find any point P(λ, -λ, λ) on AB
- Find any point Q(1, -2μ - 1, μ) on CD
- Find direction ratio's of PQ as (λ - 1, - λ + 2μ + 1, λ - μ)
- PQ ⊥ CD so use perpendicularity condition to find eqn. (1) as 3λ - 3μ = 2
- PQ ⊥ AB so use perpendicularity condition to find eqn. (2) as 3λ - 5μ = 2
- Solve eqn (1) and eqn (2) and find λ = 2/3 and μ = 0
- Find the coordinates of point P by putting the value of λ as (2/3, -2/3, 2/3)
- Find the coordinates of point Q by putting the value of μ as (1, -1, 0)
- By using distance formula find |PQ| =
Question 19
An insect is crawling along the line and another insect is crawling along the line . At what points on the lines should they reach so that the distance between them is the shortest? Find the shortest possible distance between them.
Answer: λ = -1, μ =1 and shortest distance = 9
Question 20
The equation of line is
5x - 3 = 15y + 7 = 3 - 10z. Find the direction cosines of the line
Solution Hint
Find the LCM of the coefficients of x, y and z
Divide all the terms of the line by 15 and convert the given line in standard form.
Find the direction ratios as 6, 2, - 3
Question 21
Find the direction cosines from these direction ratios as (6/7, 2/7, -3/7)
Show that the following lines are skew lines
Solution Hint
If then lines are not intersecting.
Answer : 2/7
Question 2: Find the angle between the planes
Answer
Question 3: Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of planes x + 2y + 3z - 4 = 0 and 2x + y - z + 5 = 0
Answer : 51x + 15y - 50z + 173 = 0 , λ = -29/7
Question 4: Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the plane x + 2y + 3z = 5 and 3x + 3y + z = 0.
Ans : 7x - 8y + 3z + 25 = 0
Question 5: Find the equation of the plane through (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x -2y + 4z = 10.
Ans : 18x + 17y + 4z -49 = 0
Question 6: Find the distance of the point (1, -2, 3) from the plane x - y + z = 5 measured parallel to the line
Ans : λ = 1/7, AB = 1
Question 7: Find the distance of the point (-2, 3, -4) from the given line measured ∥ to the plane 4x + 12y - 3z + 1 = 0,
Ans : λ = 2/3 , Point is (4, 5/2, 2), Distance = 17/2
Question 8: Find the equation of the plane passing through the point P(4, 6, 2) and the point of intersection of the plane x + y - z = 8 and the line
Ans
Question 9: Find the vector equation of the plane through the line of intersection of planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Hence find whether plane thus obtained contains the line
Ans 9 : Yes
Question 10: Find the angle between the following line and plane :
and plane
Ans : 0o
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