Wednesday, October 12, 2011

CBSE TEST PAPER 10TH MATHEMATICS Distance and Section Formulae

1. Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1). 

 [Hint: let the point be P(x, y, z). given points be A(1, 2, 3) and B(3, 2, - 1). Then PA = PB] ans: x – 2z = 0

2.Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (– 4, 0, 0) is equal to 10. 

[Hint: let the point be P(x, y, z). given points be A(4, 0, 0) and B(-4, 0, 0). Then PA + PB = 10]

3.Find the coordinates of the point which divides the line segment joining the points (– 2, 3, 5) and (1, – 4, 6) in the ratio (i) 2 : 3 internally, (ii) 2 : 3 externally. 

Ans: i) (-4/5, 1/5, 27/5) ii) (-8, 17, 3)

4.Given that P (3, 2, – 4), Q (5, 4, – 6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR. 

Ans : 1:2 

5. Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8). 

[hint: any point on YZ plane is of the form (0, y, z)] ans: 2:3 

6. Find the equation of the set of the points P such that its distances from the points A (3, 4, –5) and B (– 2, 1, 4) are equal. 

[hint: using distance formula PA and PB and equate it.] ans: 10 x + 6y – 18z – 29 = 0

7. Find the coordinates of the points which trisect the line segment joining the points P (4, 2, – 6) and Q (10, –16, 6). 

[hint: points of trisection divides the line segment into three equal parts. So use the ratio 1 : 2 and 2 : 1] 
Ans : (6, -4, -2) and (8, -10, 2)

8. The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C. 

[hint: use centroid formula. Let the third vertex be C(p, q, r)] 
ans: (1, 1, 2) 

9. Three vertices of a parallelogram ABCD are A(3, – 1, 2), B (1, 2, – 4) and C (– 1, 1, 2). Find the coordinates of the fourth vertex. 
Ans: (1, - 2, 8) 

10. If the origin is the centroid of the triangle PQR with vertices P (2a, 2, 6), Q (– 4, 3b, –10) and R(8, 14, 2c), then  find the values of a, b and c. 

[hint: use centroid formula] ans: a = - 2 , b = - 16/3, c = 2

Practice:

1.A point R with x-coordinate 4 lies on the line segment joining the points P(2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R. 
[hint: let coordinate of R = (4, y, z). let R divide PQ in the ratio k:1. Using section formula, find the coordinate of R and equate its x – coordinate to 4. Solve to find value of k. then using value of k , find y and z] 
ans: (4, -2, 6)

2. A is a point on the y-axis whose ordinate is 5 and B is the point (-3, 1). Calculate the length of AB.

3. The distance between A(1, 3) and B(x, 7) is 5. Find the possible values of x.

4. P and Q have co-ordinates (-1, 2) and (6, 3) respectively. Reflect P in the x-axis to P'. Find the length of the segment P'Q.

5. Point A(2, -4) is reflected in the origin as A'. Point B(-3, 2) is reflected in x-axis at B'. Write the co-ordinates of A' and B'. Calculate the distance A'B' correct to one decimal place.

6. The center of a circle of radius 13 units is the point (3, 6). P(7, 9) is a point inside the circle. APB is a chord of the circle such that AP = PB. Calculate the length of AB.

7. A and B have co-ordinates (4, 3) and (0, 1) respectively. Find (i) the image A' of A under reflection in the y-axis.(ii) the image B' of B under reflection in the line AA'.
(iii) the length of A'B'.

8. What point (or points) on the x-axis are at a distance of 5 units from the point (5, -4)?

9. Find point (or points) which are at a distance of 10 from the point (4, 3), given that the ordinate of the point (or points) is twice the abscissa.

10. Show that the points (3, 3), (9, 0) and (12, 21) are the vertices of a right angled triangle.

11. Show that the points (0, -1), (-2, 3), (6, 7) and (8, 3) are the vertices of a rectangle.

12. The points A(0, 3), B(-2, a) and C(-1, 4) are the vertices of a right angled triangle at A, find the value of a.

13. Show by distance formula that the points (-1, -1), (2, 3) and (8, 11) are collinear.

14. Calculate the co-ordinates of the point P that divides the line joining the points A (-1, 3) and B(5, -6)
internally in the ratio 1:2.

15. Find the co-ordinates of the points of trisection of the line segment joining the points (3, -3) and (6, 9).

16. The line segment joining A(-3, 1) and B(5, -4) is a diameter of a circle whose center is C. Find the co-
ordinates of the point C.

17. The mid-point of the line joining (a, 2) and (3, 6) is (2, b). Find the values of a and b.

18. The mid-point of the line segment joining (2a, 4) and (-2, 3b) is (1, 2a +1). Find the values of a and b.

19. The center of a circle is (1, -2) and one end of a diameter is (-3, 2), find the co-ordinates of the other end.

20. Find the reflection of the point (5, -3) in the point (-1, 3).


Answers



1. 3.61 units
2. 5units
3. 4 or -2
4. 74 units
5. A'(-2, 4), B'(-3, -2);
6. 1 units  6.24 units
7. (i) (-4, 3) (ii) (0, 5)
(iii) 2 5 units
8. (2, 0) and (8, 0 )
9. (1, 2), (3, 6)
10. 67.5 sq. units
12. 1  
14. (1, 0)
15. (4, 1), (5, 5)
16. (1,-3/2)
17. a = 1, b = 4
18. a = 2, b = 2
19. (5,-6)
20. (-7, 9)

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1 comment:

J Sunil said...

Let A(1, 2), B (4, 3) and C(6, 6).
Let D (x, y) be the fourth vertex of the parallelogram ABCD.
Midpoint of AC = [(1+6)/2 , (2+6)/2] = (7/2,4)
Midpoint of BD = [(x+4)/2 , (y+3)/2
Since, the diagonals of parallelogram bisect each other at O.
∴ Mid point of BD = Mid point of AC
(7/2,4) = [(x+4)/2 , (y+3)/2
 7/2 = (x+4)/2
 x = 3
also, 4 = (y+3)/2
y = 8-3 =5

Thus, (3, 5) is the fourth vertex.

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