7 - Question bank - Coordinate Geometry - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

Class 10 : 7 - Question bank - Coordinate Geometry - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


   Co-ordinate Geometry 
 
 
1) Find the distance between two points. 
 a) (4, 7) and (-3,2)   b) (0,6) and (8,0) 
Solution: 
 a) (4, 7) and (-3, 2) 
  Let A(x
1
,y
1
) ? (4,7)  and B(x
2
,y
2
) ?(-3,2) 
Distance between the points A(4,7) and B(-3,2) is given by 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
2 2
7 2 4 ) 3 ( ? ? ? ? 
         = ? ? ? ?
2 2
5 7 ? ? ? 
         =  49 25 ? 
         =  74    
  
 
Distance between the points A(4,7) and B(-3,2) is 74 units. 
 
 b) (0,6) and (8,0) 
  P(x
1
,y
1
) ? (0,6) Q(x
2
,y
2
) ?(8,0) 
  Distance between the points P(0,6) and Q(8,0) is given by 
  PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(8 0 0 6 ? ? ?
 
  
= ? ? ? ?
22
86 ??
 
  
=  64 36 ?
 
  
=  100
  
       = 10  
 
Distance between the points A(0,6) and Q(8,0) is 10 units. 
 
 
2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear.  
 
Solution: 
 Consider, D(x
1
,y
1
)
= 
(3,-1), E(x
2
,y
2
)
= 
(0,5) & F(x
3
,y
3
)=(-2,-8). 
Distance between the points D (3,-1) and E (0,5) is given by 
Page 2


   Co-ordinate Geometry 
 
 
1) Find the distance between two points. 
 a) (4, 7) and (-3,2)   b) (0,6) and (8,0) 
Solution: 
 a) (4, 7) and (-3, 2) 
  Let A(x
1
,y
1
) ? (4,7)  and B(x
2
,y
2
) ?(-3,2) 
Distance between the points A(4,7) and B(-3,2) is given by 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
2 2
7 2 4 ) 3 ( ? ? ? ? 
         = ? ? ? ?
2 2
5 7 ? ? ? 
         =  49 25 ? 
         =  74    
  
 
Distance between the points A(4,7) and B(-3,2) is 74 units. 
 
 b) (0,6) and (8,0) 
  P(x
1
,y
1
) ? (0,6) Q(x
2
,y
2
) ?(8,0) 
  Distance between the points P(0,6) and Q(8,0) is given by 
  PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(8 0 0 6 ? ? ?
 
  
= ? ? ? ?
22
86 ??
 
  
=  64 36 ?
 
  
=  100
  
       = 10  
 
Distance between the points A(0,6) and Q(8,0) is 10 units. 
 
 
2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear.  
 
Solution: 
 Consider, D(x
1
,y
1
)
= 
(3,-1), E(x
2
,y
2
)
= 
(0,5) & F(x
3
,y
3
)=(-2,-8). 
Distance between the points D (3,-1) and E (0,5) is given by 
 Co-ordinate Geometry 
 
 
 
           DE   = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(0 3 5 ( 1) ? ? ? ?
 
  
= ? ? ? ?
22
36 ??
 
  
=  9 36 ?
 
          DE    
=  45
 
Distance between the points E(0,5) and F(-2,-8) is given by 
EF   = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
  
= ? ? ? ?
22
( 2) 0 ( 8) 5 ? ? ? ? ?
 
  
= ? ? ? ?
22
2 13 ? ? ?
 
  
=  4 169 ?
 
EF    =  
173
 
Distance between the points D(3,-1) and F(-2,-8) is given by 
DF   = 
? ? ? ?
22
3 1 3 1
x x y y ? ? ?
 
          = 
? ? ? ?
22
( 2) 3 ( 8) ( 1) ? ? ? ? ? ?
 
          = 
? ? ? ?
22
57 ? ? ?
 
          =  
25 49 ?
 
          DF    =  
74
 
 
DE+EF= 45 74 ? ? 173 =DF 
Hence D, E and F are not collinear.  
 
3)  Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. 
Solution: 
 Consider, A(x
1
,y
1
)
= 
(5,4),  B(x
2
,y
2
)
= 
(6,9)  C(x
3
,y
3
)=(11,10) 
 Distance between the points A(5,4) and B(6,9) is 
Page 3


   Co-ordinate Geometry 
 
 
1) Find the distance between two points. 
 a) (4, 7) and (-3,2)   b) (0,6) and (8,0) 
Solution: 
 a) (4, 7) and (-3, 2) 
  Let A(x
1
,y
1
) ? (4,7)  and B(x
2
,y
2
) ?(-3,2) 
Distance between the points A(4,7) and B(-3,2) is given by 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
2 2
7 2 4 ) 3 ( ? ? ? ? 
         = ? ? ? ?
2 2
5 7 ? ? ? 
         =  49 25 ? 
         =  74    
  
 
Distance between the points A(4,7) and B(-3,2) is 74 units. 
 
 b) (0,6) and (8,0) 
  P(x
1
,y
1
) ? (0,6) Q(x
2
,y
2
) ?(8,0) 
  Distance between the points P(0,6) and Q(8,0) is given by 
  PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(8 0 0 6 ? ? ?
 
  
= ? ? ? ?
22
86 ??
 
  
=  64 36 ?
 
  
=  100
  
       = 10  
 
Distance between the points A(0,6) and Q(8,0) is 10 units. 
 
 
2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear.  
 
Solution: 
 Consider, D(x
1
,y
1
)
= 
(3,-1), E(x
2
,y
2
)
= 
(0,5) & F(x
3
,y
3
)=(-2,-8). 
Distance between the points D (3,-1) and E (0,5) is given by 
 Co-ordinate Geometry 
 
 
 
           DE   = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(0 3 5 ( 1) ? ? ? ?
 
  
= ? ? ? ?
22
36 ??
 
  
=  9 36 ?
 
          DE    
=  45
 
Distance between the points E(0,5) and F(-2,-8) is given by 
EF   = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
  
= ? ? ? ?
22
( 2) 0 ( 8) 5 ? ? ? ? ?
 
  
= ? ? ? ?
22
2 13 ? ? ?
 
  
=  4 169 ?
 
EF    =  
173
 
Distance between the points D(3,-1) and F(-2,-8) is given by 
DF   = 
? ? ? ?
22
3 1 3 1
x x y y ? ? ?
 
          = 
? ? ? ?
22
( 2) 3 ( 8) ( 1) ? ? ? ? ? ?
 
          = 
? ? ? ?
22
57 ? ? ?
 
          =  
25 49 ?
 
          DF    =  
74
 
 
DE+EF= 45 74 ? ? 173 =DF 
Hence D, E and F are not collinear.  
 
3)  Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. 
Solution: 
 Consider, A(x
1
,y
1
)
= 
(5,4),  B(x
2
,y
2
)
= 
(6,9)  C(x
3
,y
3
)=(11,10) 
 Distance between the points A(5,4) and B(6,9) is 
  Co-ordinate Geometry 
 
 
 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
         
=  ? ? ? ?
22
6 5 9 4 ? ? ? 
         =  ? ? ? ?
22
15 ? 
         =  1 25 ? 
  AB =  26 
 Distance between the points B (6,9) and C (11,10) is 
  BC  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
         
 = 
? ? ? ?
22
11 6 10 9 ? ? ?
  
          =  ? ? ? ?
22
51 ? 
          =   25 1 ? 
          =  26 
 Therefore, AB=BC 
 Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle.  
 
4)  What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form.  
Solution: 
 Let P(x
1
,y
1
)
= 
(-2, -1), Q(x
2
,y
2
)
= 
(2,0),  R(x
3
,y
3
) 
= 
(-2,1) &S(x
4
,y
4
) 
= 
(-3,0). 
 Distance between the points P(-2,-1) and Q(2,0) is  
   PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
22
2 ( 2) 0 ( 1) ? ? ? ? ? 
         =  ? ? ? ?
22
41 ? 
         =  16 1 ? 
          PQ   =  17   
 Distance between the points Q(2,0) and R(-2,1) is  
  QR  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ? 
                                                                          
Page 4


   Co-ordinate Geometry 
 
 
1) Find the distance between two points. 
 a) (4, 7) and (-3,2)   b) (0,6) and (8,0) 
Solution: 
 a) (4, 7) and (-3, 2) 
  Let A(x
1
,y
1
) ? (4,7)  and B(x
2
,y
2
) ?(-3,2) 
Distance between the points A(4,7) and B(-3,2) is given by 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
2 2
7 2 4 ) 3 ( ? ? ? ? 
         = ? ? ? ?
2 2
5 7 ? ? ? 
         =  49 25 ? 
         =  74    
  
 
Distance between the points A(4,7) and B(-3,2) is 74 units. 
 
 b) (0,6) and (8,0) 
  P(x
1
,y
1
) ? (0,6) Q(x
2
,y
2
) ?(8,0) 
  Distance between the points P(0,6) and Q(8,0) is given by 
  PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(8 0 0 6 ? ? ?
 
  
= ? ? ? ?
22
86 ??
 
  
=  64 36 ?
 
  
=  100
  
       = 10  
 
Distance between the points A(0,6) and Q(8,0) is 10 units. 
 
 
2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear.  
 
Solution: 
 Consider, D(x
1
,y
1
)
= 
(3,-1), E(x
2
,y
2
)
= 
(0,5) & F(x
3
,y
3
)=(-2,-8). 
Distance between the points D (3,-1) and E (0,5) is given by 
 Co-ordinate Geometry 
 
 
 
           DE   = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(0 3 5 ( 1) ? ? ? ?
 
  
= ? ? ? ?
22
36 ??
 
  
=  9 36 ?
 
          DE    
=  45
 
Distance between the points E(0,5) and F(-2,-8) is given by 
EF   = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
  
= ? ? ? ?
22
( 2) 0 ( 8) 5 ? ? ? ? ?
 
  
= ? ? ? ?
22
2 13 ? ? ?
 
  
=  4 169 ?
 
EF    =  
173
 
Distance between the points D(3,-1) and F(-2,-8) is given by 
DF   = 
? ? ? ?
22
3 1 3 1
x x y y ? ? ?
 
          = 
? ? ? ?
22
( 2) 3 ( 8) ( 1) ? ? ? ? ? ?
 
          = 
? ? ? ?
22
57 ? ? ?
 
          =  
25 49 ?
 
          DF    =  
74
 
 
DE+EF= 45 74 ? ? 173 =DF 
Hence D, E and F are not collinear.  
 
3)  Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. 
Solution: 
 Consider, A(x
1
,y
1
)
= 
(5,4),  B(x
2
,y
2
)
= 
(6,9)  C(x
3
,y
3
)=(11,10) 
 Distance between the points A(5,4) and B(6,9) is 
  Co-ordinate Geometry 
 
 
 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
         
=  ? ? ? ?
22
6 5 9 4 ? ? ? 
         =  ? ? ? ?
22
15 ? 
         =  1 25 ? 
  AB =  26 
 Distance between the points B (6,9) and C (11,10) is 
  BC  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
         
 = 
? ? ? ?
22
11 6 10 9 ? ? ?
  
          =  ? ? ? ?
22
51 ? 
          =   25 1 ? 
          =  26 
 Therefore, AB=BC 
 Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle.  
 
4)  What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form.  
Solution: 
 Let P(x
1
,y
1
)
= 
(-2, -1), Q(x
2
,y
2
)
= 
(2,0),  R(x
3
,y
3
) 
= 
(-2,1) &S(x
4
,y
4
) 
= 
(-3,0). 
 Distance between the points P(-2,-1) and Q(2,0) is  
   PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
22
2 ( 2) 0 ( 1) ? ? ? ? ? 
         =  ? ? ? ?
22
41 ? 
         =  16 1 ? 
          PQ   =  17   
 Distance between the points Q(2,0) and R(-2,1) is  
  QR  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ? 
                                                                          
  Co-ordinate Geometry 
 
 
 
 =   ? ? ? ?
22
( 2) 2 1 0 ? ? ? ? 
  =   
? ? ? ?
22
41 ??   
 = 16 1 ? 
 QR  = 17 
 Distance between the points R (-2,1) and S (-3,0) is 
   RS  = ? ? ? ?
22
4 3 4 3
x x y y ? ? ? 
          =   ? ? ? ?
22
( 3) ( 2) 0 1 ? ? ? ? ? 
          =   ? ? ? ?
22
11 ? ? ? 
  = 11 ? 
               RS =   2 
 Distance between the points S(-3,0) and P(-2,-1) is 
  SP = ? ? ? ?
22
4 1 4 1
x x y y ? ? ?  
  =   ? ? ? ?
22
( 2) ( 3) ( 1) 0 ? ? ? ? ? ? 
  =  ? ? ? ?
22
11 ?? 
  =  11 ? 
               SP =  2 
  
 Therefore PQ=QR and RS=SP 
 Thus Opposites sides of the quadrilateral PQRS are equal.  
 Therefore by definition, 
  
PQRS is a parallelogram 
 
5) Show that the points (1,7) (4,2) (-1,-1) and (-4,4) form the vertices of a square.  
Solution:  
 Consider, A(x
1
,y
1
) ? (1,7),  B(x
2
,y
2
) ?(4,2),  C(x
3
,y
3
) ? (-1, -1)& D(x
4
,y
4
) ? (-4,4) 
 Distance between the points A(1,7) and B(4,2) is  
            AB  =  ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
                          =    ? ? ? ?
22
4 1 2 7 ? ? ? 
Page 5


   Co-ordinate Geometry 
 
 
1) Find the distance between two points. 
 a) (4, 7) and (-3,2)   b) (0,6) and (8,0) 
Solution: 
 a) (4, 7) and (-3, 2) 
  Let A(x
1
,y
1
) ? (4,7)  and B(x
2
,y
2
) ?(-3,2) 
Distance between the points A(4,7) and B(-3,2) is given by 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
2 2
7 2 4 ) 3 ( ? ? ? ? 
         = ? ? ? ?
2 2
5 7 ? ? ? 
         =  49 25 ? 
         =  74    
  
 
Distance between the points A(4,7) and B(-3,2) is 74 units. 
 
 b) (0,6) and (8,0) 
  P(x
1
,y
1
) ? (0,6) Q(x
2
,y
2
) ?(8,0) 
  Distance between the points P(0,6) and Q(8,0) is given by 
  PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(8 0 0 6 ? ? ?
 
  
= ? ? ? ?
22
86 ??
 
  
=  64 36 ?
 
  
=  100
  
       = 10  
 
Distance between the points A(0,6) and Q(8,0) is 10 units. 
 
 
2) Determine if the points (3,-1), (0,5) and (-2,-8) are collinear.  
 
Solution: 
 Consider, D(x
1
,y
1
)
= 
(3,-1), E(x
2
,y
2
)
= 
(0,5) & F(x
3
,y
3
)=(-2,-8). 
Distance between the points D (3,-1) and E (0,5) is given by 
 Co-ordinate Geometry 
 
 
 
           DE   = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
  
= ? ? ? ?
22
(0 3 5 ( 1) ? ? ? ?
 
  
= ? ? ? ?
22
36 ??
 
  
=  9 36 ?
 
          DE    
=  45
 
Distance between the points E(0,5) and F(-2,-8) is given by 
EF   = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
  
= ? ? ? ?
22
( 2) 0 ( 8) 5 ? ? ? ? ?
 
  
= ? ? ? ?
22
2 13 ? ? ?
 
  
=  4 169 ?
 
EF    =  
173
 
Distance between the points D(3,-1) and F(-2,-8) is given by 
DF   = 
? ? ? ?
22
3 1 3 1
x x y y ? ? ?
 
          = 
? ? ? ?
22
( 2) 3 ( 8) ( 1) ? ? ? ? ? ?
 
          = 
? ? ? ?
22
57 ? ? ?
 
          =  
25 49 ?
 
          DF    =  
74
 
 
DE+EF= 45 74 ? ? 173 =DF 
Hence D, E and F are not collinear.  
 
3)  Check whether (5,4) (6,9) and (11,10) form the vertices of an isosceles triangle. 
Solution: 
 Consider, A(x
1
,y
1
)
= 
(5,4),  B(x
2
,y
2
)
= 
(6,9)  C(x
3
,y
3
)=(11,10) 
 Distance between the points A(5,4) and B(6,9) is 
  Co-ordinate Geometry 
 
 
 
  AB = ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
         
=  ? ? ? ?
22
6 5 9 4 ? ? ? 
         =  ? ? ? ?
22
15 ? 
         =  1 25 ? 
  AB =  26 
 Distance between the points B (6,9) and C (11,10) is 
  BC  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?
 
         
 = 
? ? ? ?
22
11 6 10 9 ? ? ?
  
          =  ? ? ? ?
22
51 ? 
          =   25 1 ? 
          =  26 
 Therefore, AB=BC 
 Hence A (5,4), B (6,9) and C (11,10) form the vertices of an isoscelestriangle.  
 
4)  What type of quadrilaterals to the points (-2,-1) (2,0) (-2,1) and (-3,0) form.  
Solution: 
 Let P(x
1
,y
1
)
= 
(-2, -1), Q(x
2
,y
2
)
= 
(2,0),  R(x
3
,y
3
) 
= 
(-2,1) &S(x
4
,y
4
) 
= 
(-3,0). 
 Distance between the points P(-2,-1) and Q(2,0) is  
   PQ = ? ? ? ?
22
2 1 2 1
x x y y ? ? ? 
         = ? ? ? ?
22
2 ( 2) 0 ( 1) ? ? ? ? ? 
         =  ? ? ? ?
22
41 ? 
         =  16 1 ? 
          PQ   =  17   
 Distance between the points Q(2,0) and R(-2,1) is  
  QR  = ? ? ? ?
22
3 2 3 2
x x y y ? ? ? 
                                                                          
  Co-ordinate Geometry 
 
 
 
 =   ? ? ? ?
22
( 2) 2 1 0 ? ? ? ? 
  =   
? ? ? ?
22
41 ??   
 = 16 1 ? 
 QR  = 17 
 Distance between the points R (-2,1) and S (-3,0) is 
   RS  = ? ? ? ?
22
4 3 4 3
x x y y ? ? ? 
          =   ? ? ? ?
22
( 3) ( 2) 0 1 ? ? ? ? ? 
          =   ? ? ? ?
22
11 ? ? ? 
  = 11 ? 
               RS =   2 
 Distance between the points S(-3,0) and P(-2,-1) is 
  SP = ? ? ? ?
22
4 1 4 1
x x y y ? ? ?  
  =   ? ? ? ?
22
( 2) ( 3) ( 1) 0 ? ? ? ? ? ? 
  =  ? ? ? ?
22
11 ?? 
  =  11 ? 
               SP =  2 
  
 Therefore PQ=QR and RS=SP 
 Thus Opposites sides of the quadrilateral PQRS are equal.  
 Therefore by definition, 
  
PQRS is a parallelogram 
 
5) Show that the points (1,7) (4,2) (-1,-1) and (-4,4) form the vertices of a square.  
Solution:  
 Consider, A(x
1
,y
1
) ? (1,7),  B(x
2
,y
2
) ?(4,2),  C(x
3
,y
3
) ? (-1, -1)& D(x
4
,y
4
) ? (-4,4) 
 Distance between the points A(1,7) and B(4,2) is  
            AB  =  ? ? ? ?
22
2 1 2 1
x x y y ? ? ?
 
                          =    ? ? ? ?
22
4 1 2 7 ? ? ? 
Co-ordinate Geometry 
 
 
 
 
 
  =    ? ? ? ?
22
35 ?? 
  =   9 25 ? 
            AB   =    34 units.................... (1) 
      Distance between the points B(4,2) and C(-1,-1) is  
             BC   = ? ? ? ?
22
3 2 3 2
x x y y ? ? ?   
  =   ? ? ? ?
22
( 1) 4 ( 1) 2 ? ? ? ? ? 
 =   ? ? ? ?
22
53 ? ? ? 
 =  25 9 ? 
                 BC =  34 units................. (2)     
         Distance between the points C(-1,-1) and D(-4,4) is 
             CD = ? ? ? ?
22
4 3 4 3
x x y y ? ? ?
 
  =  ? ? ? ?
22
( 4) ( 1) 4 ( 1) ? ? ? ? ? ? 
  =   ? ? ? ?
22
35 ?? 
  =  9 25 ? 
             CD =   34 units........................... (3) 
        Distance between the points D (-4,4) and A (1,7) is 
             DA =  ? ? ? ?
22
1 4 1 4
x x y y ? ? ? 
  =  ? ? ? ?
22
1 ( 4) 7 4 ? ? ? ? 
  =  ? ? ? ?
22
53 ? 
  =  25 9 ? 
                   DA =  34 units.......................(4)  
 Distance between the points A(1,7) and C(-1, -1) is 
              AC = ? ? ? ?
22
3 1 3 1
x x y y ? ? ?  
  =  ? ? ? ?
22
( 1) 1 ( 1) 7 ? ? ? ? ? 
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7 - Question bank - Coordinate Geometry - Class 10 - Maths Class 10 Notes | EduRev

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7 - Question bank - Coordinate Geometry - Class 10 - Maths Class 10 Notes | EduRev

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7 - Question bank - Coordinate Geometry - Class 10 - Maths Class 10 Notes | EduRev

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