Page 1
1. Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
Sol:
The given points are 1, , 8 5,7 A y and 2, 3 . O y
Here, AO and BO are the radii of the circle. So
2 2
AO BO AO BO
2 2 2 2
2 2 2
2 2
2
2
2
2 1 3 2 5 3 7
9 4 3 3 7
9 16 9 9 49 42
7 42 49 0
6 7 0
7 7 0
7 1 7 0
7 1 0
1 7
y y y
y y
y y y
y y
y y
y y y
y y y
y y
y or y
Hence, 7 1. y or y
2. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Sol:
The given ports are 0,2 , 3, A B p and ,5 . C p
Page 2
1. Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
Sol:
The given points are 1, , 8 5,7 A y and 2, 3 . O y
Here, AO and BO are the radii of the circle. So
2 2
AO BO AO BO
2 2 2 2
2 2 2
2 2
2
2
2
2 1 3 2 5 3 7
9 4 3 3 7
9 16 9 9 49 42
7 42 49 0
6 7 0
7 7 0
7 1 7 0
7 1 0
1 7
y y y
y y
y y y
y y
y y
y y y
y y y
y y
y or y
Hence, 7 1. y or y
2. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Sol:
The given ports are 0,2 , 3, A B p and ,5 . C p
2 2
2 2 2 2
2 2
3 0 2 0 5 2
9 4 4 9
4 4 1
AB AC AB AC
p p
p p p
p p
Hence, 1. p
3. ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of
one its diagonal.
Sol:
The given vertices are B(4, 0), C(4, 3) and D(0, 3) Here, BD one of the diagonals So
2 2
2 2
4 0 0 3
4 3
16 9
BD
25
5
Hence, the length of the diagonal is 5 units.
4. If the point P(k-1, 2) is equidistant from the points A(3,k) and B(k,5), find the value of k.
Sol:
The given points are 1 ,2 , 3, ,5 . P k A k and B k
2 2
AP BP
AP BP
2 2 2 2
2 2 2 2
2 2
2
1 3 2 1 2 5
4 2 1 3
8 16 4 4 1 9
6 5 0
1 5 0
1 5
k k k k
k k
k y k k
k y
k k
k or k
Hence, 1 5 k or k
5. Find the ratio in which the point P(x,2) divides the join of A(12, 5) and B(4, -3).
Sol:
Let k be the ratio in which the point ,2 P x divides the line joining the points
1 1 2 2
12, 5 4, 3 . A x y and B x y Then
Page 3
1. Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
Sol:
The given points are 1, , 8 5,7 A y and 2, 3 . O y
Here, AO and BO are the radii of the circle. So
2 2
AO BO AO BO
2 2 2 2
2 2 2
2 2
2
2
2
2 1 3 2 5 3 7
9 4 3 3 7
9 16 9 9 49 42
7 42 49 0
6 7 0
7 7 0
7 1 7 0
7 1 0
1 7
y y y
y y
y y y
y y
y y
y y y
y y y
y y
y or y
Hence, 7 1. y or y
2. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Sol:
The given ports are 0,2 , 3, A B p and ,5 . C p
2 2
2 2 2 2
2 2
3 0 2 0 5 2
9 4 4 9
4 4 1
AB AC AB AC
p p
p p p
p p
Hence, 1. p
3. ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of
one its diagonal.
Sol:
The given vertices are B(4, 0), C(4, 3) and D(0, 3) Here, BD one of the diagonals So
2 2
2 2
4 0 0 3
4 3
16 9
BD
25
5
Hence, the length of the diagonal is 5 units.
4. If the point P(k-1, 2) is equidistant from the points A(3,k) and B(k,5), find the value of k.
Sol:
The given points are 1 ,2 , 3, ,5 . P k A k and B k
2 2
AP BP
AP BP
2 2 2 2
2 2 2 2
2 2
2
1 3 2 1 2 5
4 2 1 3
8 16 4 4 1 9
6 5 0
1 5 0
1 5
k k k k
k k
k y k k
k y
k k
k or k
Hence, 1 5 k or k
5. Find the ratio in which the point P(x,2) divides the join of A(12, 5) and B(4, -3).
Sol:
Let k be the ratio in which the point ,2 P x divides the line joining the points
1 1 2 2
12, 5 4, 3 . A x y and B x y Then
3 5
4 12
2
1 1
k
k
x and
k k
Now,
3 5
3
2 2 2 3 5
1 5
k
k k k
k
Hence, the required ratio is 3: 5.
6. Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and
D(2,6) are equal and bisect each other.
Sol:
The vertices of the rectangle ABCD are 2, 1 , 5, 1 5,6 , A B C and 2,6 . D Now
Coordinates of midpoint of
2 5 1 6 7 5
, ,
2 2 2 2
AC
Coordinates of midpoint of
5 2 1 6 7 5
, ,
2 2 2 2
BD
Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD
bisect each other
7. Find the lengths of the medians AD and BE of ABC whose vertices are A(7,-3), B(5,3)
and C(3,-1)
Sol:
The given vertices are 7, 3 , 5,3 3, 1 . - A B and C
Since D and E are the midpoints of BC and AC respectively. therefore
Coordinates of
5 3 3 1
, 4,1
2 2
D
Coordinates of
7 3 3 1
, 5, 2
2 2
E
Now
2 2
2 2
7 4 3 1 9 16 5
5 5 3 2 0 25 5
AD
BE
Hence, AD = BE = 5 units.
8. If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value
of k.
Sol:
Here, the point ,4 C k divides the join of 2,6 A and 5,1 B in ratio 2 : 3. So
Page 4
1. Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
Sol:
The given points are 1, , 8 5,7 A y and 2, 3 . O y
Here, AO and BO are the radii of the circle. So
2 2
AO BO AO BO
2 2 2 2
2 2 2
2 2
2
2
2
2 1 3 2 5 3 7
9 4 3 3 7
9 16 9 9 49 42
7 42 49 0
6 7 0
7 7 0
7 1 7 0
7 1 0
1 7
y y y
y y
y y y
y y
y y
y y y
y y y
y y
y or y
Hence, 7 1. y or y
2. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Sol:
The given ports are 0,2 , 3, A B p and ,5 . C p
2 2
2 2 2 2
2 2
3 0 2 0 5 2
9 4 4 9
4 4 1
AB AC AB AC
p p
p p p
p p
Hence, 1. p
3. ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of
one its diagonal.
Sol:
The given vertices are B(4, 0), C(4, 3) and D(0, 3) Here, BD one of the diagonals So
2 2
2 2
4 0 0 3
4 3
16 9
BD
25
5
Hence, the length of the diagonal is 5 units.
4. If the point P(k-1, 2) is equidistant from the points A(3,k) and B(k,5), find the value of k.
Sol:
The given points are 1 ,2 , 3, ,5 . P k A k and B k
2 2
AP BP
AP BP
2 2 2 2
2 2 2 2
2 2
2
1 3 2 1 2 5
4 2 1 3
8 16 4 4 1 9
6 5 0
1 5 0
1 5
k k k k
k k
k y k k
k y
k k
k or k
Hence, 1 5 k or k
5. Find the ratio in which the point P(x,2) divides the join of A(12, 5) and B(4, -3).
Sol:
Let k be the ratio in which the point ,2 P x divides the line joining the points
1 1 2 2
12, 5 4, 3 . A x y and B x y Then
3 5
4 12
2
1 1
k
k
x and
k k
Now,
3 5
3
2 2 2 3 5
1 5
k
k k k
k
Hence, the required ratio is 3: 5.
6. Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and
D(2,6) are equal and bisect each other.
Sol:
The vertices of the rectangle ABCD are 2, 1 , 5, 1 5,6 , A B C and 2,6 . D Now
Coordinates of midpoint of
2 5 1 6 7 5
, ,
2 2 2 2
AC
Coordinates of midpoint of
5 2 1 6 7 5
, ,
2 2 2 2
BD
Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD
bisect each other
7. Find the lengths of the medians AD and BE of ABC whose vertices are A(7,-3), B(5,3)
and C(3,-1)
Sol:
The given vertices are 7, 3 , 5,3 3, 1 . - A B and C
Since D and E are the midpoints of BC and AC respectively. therefore
Coordinates of
5 3 3 1
, 4,1
2 2
D
Coordinates of
7 3 3 1
, 5, 2
2 2
E
Now
2 2
2 2
7 4 3 1 9 16 5
5 5 3 2 0 25 5
AD
BE
Hence, AD = BE = 5 units.
8. If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value
of k.
Sol:
Here, the point ,4 C k divides the join of 2,6 A and 5,1 B in ratio 2 : 3. So
2 5 3 2
2 3
10 6
5
16
5
k
Hence,
16
.
5
k
9. Find the point on x-axis which is equidistant from points A(-1,0) and B(5,0)
Sol:
Let ,0 P x be the point on . x axis Then
2 2
2 2 2 2
2 2
1 0 0 5 0 0
2 1 10 25
12 24 2
AP BP AP BP
x x
x x x x
x x
Hence, 2 x
10. Find the distance between the points
8
,2
5
A and
2
,2
5
B
Sol:
The given points are
8
,2
5
A and
2
,2
5
B
Then,
1 1 2 2
8 2
, 2 , 2
5 5
x y and x y
Therefore,
2 2
2 1 2 1
2
2
2 2
2 8
2 2
5 5
2 0
4 0
4
2 .
AB x x y y
units
Page 5
1. Points A(-1, y) and B(5,7) lie on the circle with centre O(2, -3y).Find the value of y.
Sol:
The given points are 1, , 8 5,7 A y and 2, 3 . O y
Here, AO and BO are the radii of the circle. So
2 2
AO BO AO BO
2 2 2 2
2 2 2
2 2
2
2
2
2 1 3 2 5 3 7
9 4 3 3 7
9 16 9 9 49 42
7 42 49 0
6 7 0
7 7 0
7 1 7 0
7 1 0
1 7
y y y
y y
y y y
y y
y y
y y y
y y y
y y
y or y
Hence, 7 1. y or y
2. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find p.
Sol:
The given ports are 0,2 , 3, A B p and ,5 . C p
2 2
2 2 2 2
2 2
3 0 2 0 5 2
9 4 4 9
4 4 1
AB AC AB AC
p p
p p p
p p
Hence, 1. p
3. ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of
one its diagonal.
Sol:
The given vertices are B(4, 0), C(4, 3) and D(0, 3) Here, BD one of the diagonals So
2 2
2 2
4 0 0 3
4 3
16 9
BD
25
5
Hence, the length of the diagonal is 5 units.
4. If the point P(k-1, 2) is equidistant from the points A(3,k) and B(k,5), find the value of k.
Sol:
The given points are 1 ,2 , 3, ,5 . P k A k and B k
2 2
AP BP
AP BP
2 2 2 2
2 2 2 2
2 2
2
1 3 2 1 2 5
4 2 1 3
8 16 4 4 1 9
6 5 0
1 5 0
1 5
k k k k
k k
k y k k
k y
k k
k or k
Hence, 1 5 k or k
5. Find the ratio in which the point P(x,2) divides the join of A(12, 5) and B(4, -3).
Sol:
Let k be the ratio in which the point ,2 P x divides the line joining the points
1 1 2 2
12, 5 4, 3 . A x y and B x y Then
3 5
4 12
2
1 1
k
k
x and
k k
Now,
3 5
3
2 2 2 3 5
1 5
k
k k k
k
Hence, the required ratio is 3: 5.
6. Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and
D(2,6) are equal and bisect each other.
Sol:
The vertices of the rectangle ABCD are 2, 1 , 5, 1 5,6 , A B C and 2,6 . D Now
Coordinates of midpoint of
2 5 1 6 7 5
, ,
2 2 2 2
AC
Coordinates of midpoint of
5 2 1 6 7 5
, ,
2 2 2 2
BD
Since, the midpoints of AC and BD coincide, therefore the diagonals of rectangle ABCD
bisect each other
7. Find the lengths of the medians AD and BE of ABC whose vertices are A(7,-3), B(5,3)
and C(3,-1)
Sol:
The given vertices are 7, 3 , 5,3 3, 1 . - A B and C
Since D and E are the midpoints of BC and AC respectively. therefore
Coordinates of
5 3 3 1
, 4,1
2 2
D
Coordinates of
7 3 3 1
, 5, 2
2 2
E
Now
2 2
2 2
7 4 3 1 9 16 5
5 5 3 2 0 25 5
AD
BE
Hence, AD = BE = 5 units.
8. If the point C(k,4) divides the join of A(2,6) and B(5,1) in the ratio 2:3 then find the value
of k.
Sol:
Here, the point ,4 C k divides the join of 2,6 A and 5,1 B in ratio 2 : 3. So
2 5 3 2
2 3
10 6
5
16
5
k
Hence,
16
.
5
k
9. Find the point on x-axis which is equidistant from points A(-1,0) and B(5,0)
Sol:
Let ,0 P x be the point on . x axis Then
2 2
2 2 2 2
2 2
1 0 0 5 0 0
2 1 10 25
12 24 2
AP BP AP BP
x x
x x x x
x x
Hence, 2 x
10. Find the distance between the points
8
,2
5
A and
2
,2
5
B
Sol:
The given points are
8
,2
5
A and
2
,2
5
B
Then,
1 1 2 2
8 2
, 2 , 2
5 5
x y and x y
Therefore,
2 2
2 1 2 1
2
2
2 2
2 8
2 2
5 5
2 0
4 0
4
2 .
AB x x y y
units
11. Find the value of a, so that the point 3,a lies on the line represented by 2 3 5. x y
Sol:
The points 3,a lies on the line 2 3 5. x y
If point 3,a lies on the line 2 3 5, x y then 2 3 5 x y
2 3 3 5
6 3 5
3 1
1
3
a
a
a
a
Hence, the value of a is
1
.
3
12. If the points 4,3 ,5 A and B x lie on the circle with center 2,3 , O find the value of x.
Sol:
The given points A(4, 3) and B(x, 5) lie on the circle with center O(2, 3).
Then, OA = OB
2 2 2 2
2 5 3 4 2 3 3 x
2
2 2 2
2
2 2
2 2 2 0
2 2 2
x
x
2
2 0
2 0
2
x
x
x
Hence, the value of 2 x
13. If , P x y is equidistant from the points 7,1 3,5 , A and B find the relation between x
and y.
Sol:
Let the point P(x, y) be equidistant from the points A(7, 1) and B(3, 5)
Then,
2 2
2 2 2 2
7 1 3 5
PA PB
PA PB
x y x y
2 2 2 2
14 2 50 6 10 34 x y x y x y x y
8 8 16 x y
2 x y
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