RAJ KUMAR SAHU

CONTACT: 9810019908,9212708930

On the x-axis, y=0 :: On the y-axis, x=0

X →abscissa, y→ordinate
A B
(x,y) (x2,y2)
2
AB= √(𝑥2 − 𝑥1 )2 − (𝑦2−𝑦1 )
Mid point:
𝑥 +𝑥 𝑦1 +𝑦2
x= 1 2 y=
2 2
Section Formula
m1 : m2
A C B
(x1,y1) (x,y) (x2,y2)
𝑚1 𝑥2 +𝑚2 𝑥1 𝑚1 𝑥2 +𝑚2 𝑥1
x= , y=
2 2
For collinear
AB+𝐵𝐶 =AC
Equilateral triangle
√3 2
Area= a
4
AB= BC = CA
Right angle triangle.
AB2+𝐵𝐶 2 = 𝐴𝐶 2
Square:
AB= 𝐵𝐶 = 𝐷𝐶 = 𝐴𝐷
And AC= 𝐵𝐷

Rectangle:
AB= 𝐷𝐶
AD=BC
And AC=BD

Parallelogram:
AB=DC, AD=BC
And AC=BD

Rhombus:
AB=DC= 𝐴𝐷 =BC
And AC=BD
Triangle

1
Area = [𝑥1 (𝑦2 − 𝑦3 ) + 𝑥2 (𝑦3 − 𝑦1 ) + 𝑥3 (𝑦1 − 𝑦2 )]
2

Centroid:
𝑥 +𝑥 +𝑥 𝑦1 +𝑦2 +𝑦3 𝑧1 +𝑧2 +𝑧3
x= 1 2 3 y= z=
3 3 3
QUESTIONS:
Q1. Write the axis on which the given point lies
a. (2,0) e. (0,5)
b. (0, −5) f. (−2,0)
c. (−4,0) g. (12,0)
d. (0, −1) h. (0, −5)
Q2. For each of the following point write the quadrant in which it lies:
a. (−6,3) e. (2,7)
b. (8,7) f. (5, −7)
c. (2, −7) g. (−2,7)
d. (−2, −3) h. (5, −3)
Q3. The three vertices of ∆ABC are(1,4), B(−2,2) and (3,2). Plot these point on a
graph paper and calculate the area of ∆ABC.
Q4. The three vertices of square ABCD are A(3,2), B(−2,2) and D(3, −3), plot these
point on the graph paper and hence, find the coordinate as C. Also, find the area of
square ABCD.
Q5. The three vertices of rectangle ABCD are A(2,2), B(−3,2) and C ( −3,5), plot
these point on the graph paper and find the coordinate as D. Also, find the area of
rectangle ABCD.
Q6. Plot the following point on coordinate axis:
i. (2,3) vii. (3, −2)
ii. (−3,7) viii. (−3,5)
iii. (2, −3) ix. (2,0)
iv. (5,3) x. (−5,0)
v. (0,8) xi. (0, −2)
vi. (−2,7)
Q7. Write the ordinate and abscissa of following:
i. (7,3) iv. (−2,5)
ii. (2, −5) v. (3, −2)
iii. (0, −4) vi. (8,0)
Q8. Find the distance between the points:
a. A(5, −12) and B(9, −12)
b. A(1, −3) and B(4, −6)
c. A(a +𝑏, 𝑎 −b) and Q(a−b, a+b)
Q9. Find the distance of each of the following points from the origins :
a. A(5, −12)
b. B(−5,5)
c. C(−4, −6)
Q10. Find all possible values of y for which the distance between the points A(2, −3)
and B(10,y) is 10 units.
Q11. Find all possible values of x for which the distance between the points P(x, 4)
and Q(9,10) is 10 units.
Q12. If the point A(x,2) is equidistant from the points B(8, −2) and C(2, −2), find the
value of x. Also, find the length of AB.
Q13. If the point A(0,2) is equidistant from the points B(3,p) and C(p, 5), find the
value of p. Also, find the length of AB.
Q14. Find the coordinates of the point on x−axis. which is equidistant from the
point(−2,5) and (2, −3).
Q15. Find point on x−axis, each of which is at a distanceof 10 unit from the point
A(11,−8).
Q16. If the point P(x,y) is equidistance from the point A(5,1) and B(−1,5), prove that
3x= 2y.
Q17. If P(x,y) is a point equidistance from the point A(6, −1) and B(2,3), show that
x−𝑦 = 3.
Q18. If the point A(4,3) and B(x,5) lie on a circle with centre O(2,3), find the value of
x. hint: −OA2=OB2.
Q19. a. If the point (x, y) is equidistant from the point (a+𝑏, 𝑏 −a) and (a−𝑏, 𝑎 +
𝑏),prove that bx =ay.
b. If the distances of P(x, y) from A (5,1) and (−1,5)are equal then ,prove that 3x
=2y.
Q20. Using the distance formula, shows that the given points are collinear:
a. (1, −1),(5,2) and (9,5)
b. (6,9), (0,1) and (−6, −7)
Q21. Show that the point(−3, −3),(3,3) And (−3√3, 3√3)are the vertices of an
equilateral triangle.
Q22. Show that the point A(−5,6), B(3,0) and C(9,8) are the vertices of an isosceles
right−angled triangle. Calculated its area.
Q23. Show that the point O(0,0), A(3, √3) and B(3, −√3) are the vertices of an
equilateral triangle. Find the area of this triangle.
Q24. Show that the following points are vertices of a square.:
a. A(3,2), B(0,5),C(−3,2) and D(0, −1)
b. A(6,2),B(2,1),C(1,5) and D(5,6)
Q25. Show that the points A(−3,2),B(− 5, −5),C(2, −3) and D(4.4) are the vertices of
a rhombus. Find the area of the rhombus.
1
Hint:. area from rhombus = 2 × (product of its diagonal).

Q26. Show that the points A(6,1),B(8,2),C(9,4) and D(7,3) are the vertices of a
rhombus. Find its area.
Q27. Show that A(1,2), B(4,3),C(6,6) and D(3,5) are the vertices of a parallelogram. Is
this figure a rectangle?
Q28. Show that the following points are the vertices of rectangle:
a. A(−4, −2), B(−2, −4), C(4,0) and D(2,3)
b. A(2, −2), B(14,10), C(11,13) and D(−1,1)
Q29. Find the coordinates of a point P on the line segment joining A(1,2) and B(6,7)
2
such that AP = AB.
5

Q30. In what ratio does the point P(2, −5) divides the line segment joining A(−3,5)
and B(4, −9).
Q31. Find the ratio in which the point P(x,2) divides the line segment joining the
points A(12,5) and B(4, −3). Also find the value of x.
Q32. Find the length of the median AD and BE of ∆ABC whose vertices are A(7, −3)
B(5,3) & C(3, −1).
Q33. If the coordinates of points A and B are (−2, −2) and (2, −4) respectively, find
3
the coordinates of the point P such that AP = AB, where P lies on the line segment
7
AB.
Q34. Point A lies on the line segment PQ joining P(6, −6) & Q(−4, −1) in such a way
𝑃𝐴 2
that = . If the point A also lies on the line 3x+𝑘(𝑦 + 1) =0 ,find the value of k.
𝑃𝑄 5

Q35. Point P,Q and R in that order are dividing a line segment joining A(1,6) and
B(5, −2) in four equal parts. Find the coordinates of P,Q and R.
Q36. If (2,p) is the midpoint of the line segment joining the points A(6, −5) and
B(−2,11), find the value of p.
Q37. The midpoint of the line segment joining A(2a, 4) and B(−2,3b) is C(1,2a+1)
Find the value of a and b.
Q38. Find the coordinates of a point A ,where AB is a diameter of a circle with centre
C(2, −3) and the other end of the diameter is B(1,4).
3 5
Q39. Find the ratio in which the point P( , ) divides the line segment joining the
4 12
1 3
points A( , ) and B(2, −5).
2 2

Q40. Find the ratio in which the point P(m,6) divides the join of A(−4,3) and B(2,8).
Also find the value of m.
Q41. In what ratio is the line segment joining A(2, −3) and B(5,6) divided by the
x−axis? Also find the coordinate of the point of division.
Q42. In what ratio does the line x−y -2= 0 divide the line segment joining the points
A(3, −1) and B(8,9).
Q43. If G(−2,1) is the centroid of ∆ABC and two of its vertices are A(1, −6) and
B(−5,2), find the 3rd vertex of the triangle.
Q44. In what ratio does y−axis divide the line segment joining the points (−4,7)
and(3, −7).
1
Q45. If the point P( , 𝑦) lies on the line segment joining the points A(3, −5) and
2
B(−7,9), then find the ratio in which P divides AB. Also, find the value of y.
24
Q46. If what ratio does the point ( , 𝑦) divides the line segment joining the points
11
P(2, −2) and Q(3,7)? Also, find the value of y.
Q47. Multiple Choice Questions: −
1. The distance of the point P( −6,8) from the origin is:
a. 8 b. 2√7 c. 6 d. 10
2. The point on x−axis which is equidistant from point A(−1,0) and B(5,0):
a. (0,2) b. (2,0) c. (3,0) d. (0,3)
3. If R(5,6) is the midpoint of the line segment AB joining the points A(6,5) and
B(4,4), then y equals:
a. 5 b. 7 c. 12 d. 6
4. If the point C(k,4) divides the join of the points A(2,6) and B(5,1) in the ratio 2:3
then value of k is:
28 16 8
a. 16 b. c. d.
5 5 5