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Coordinate geometry Solution

Section Formula –

internal division –

Que. 1. Find the coordinates of the point which divides the line segment joining the points (-1, 7) and (4, 3) in the ratio 2 : 3.

(-1,7)                                                     (y – 3)

(x₁,y₁)                                          (x₂,y₂)

m₁ : m₂ = 2 : 3

Que. 2. Find the coordinates of the points which trisect the line segment joining the points (4, 1 ) and (-2 – 3).

m₁ : m₂ =  1 : 2      

Que. 3. To organize sports activities in your school, rows have been drawn with lime at a distance of 1m from each other on a rectangular field ABCD. 100 flower pots have been placed along AD at a distance of 1m from each other. Niharika runs a distance equal to 1/4 of AD in the second row and plants a green flag there. Preet runs a distance equal to 1/5 of AD in the eighth row and plants a red flag there. What is the distance between the two flags? If Rashmi has to plant a blue flag exactly halfway along the line joining these two flags, where should she plant her flag?

Que. 4. In what ratio does the point (-1,6) divide the line segment joining the points (3,10) and (6,-8)?

-1 (m₁ + m₂) = 6m₁ – 3m₂

-m₁ – m₂ = 6m₁ – 3m₂

-m₁ – 6m₁ =  – 3m₂ + m₂

-7 m₁ = -2 m₂

7 m₁ = 2 m₂

6 (m₁ + m₂) = -8m₁ + 10m₂

6m₁ + 6m₂ = -8m₁ + 10m₂

6m₁ + 8m₁ = 10m₂ – 6m₂

14 m₁ = 4 m₂

m₁ : m₂ =  2 : 7

Que. 5. Find the ratio in which the line segment joining the points A ( 1 – 5 ) and B ( – 4, 5 ) is divided by the x-axis. Also find the coordinates of this division point.

(1,-5)                    (x,0)                         (-4,5)

m₁ : m₂ =  K : 1

0 X K + 1 = 5K – 5

0 = 5K – 5

5 = 5K

K = 1

m₁ : m₂ =  1 : 1

Que. 6. If the points (1, 2), (4, y), (x, 6) and (3, 5), taken in that order, are the vertices of a parallelogram, then find x and y.

The diagonals of a parallelogram bisect each other, so O is the midpoint of AC.

The coordinates of point O on the AC diagonal are:

1 + x = 7

X = 7 -1

X = 6

Y + 5 = 4 X 2

Y + 5 = 8

Y = 8 – 5

Y = 3

Que. 7. Find the coordinates of point A, where AB is the diameter of a circle with centre (2, – 3 ) and coordinates of B are (1, 4).

The diameter of the circle divides it into two equal parts.

X = 3                                y₁ + 4 = -3 X 2

                                         y₁ + 4 = -6

                                         y₁ = -6 -4

                                         y₁ = -10

Que. 8. If A and B are ( – 2 – 2 ) and ( 2, 4 ) respectively, then find the coordinates of point P such that AP = 3/7 AB and P lies on the line segment AB.

(-2,-2)                    3:4                         (2,-4)

AP + BP = AB

3 + BP = 7

BP = 7 – 3

BP = 4

Que. 9. Find the coordinates of the points which divide the line segment AB joining the points A (- 2, 2) and B (2, 8) into four equal parts.

Point P divides the line segment AB. = 1 : 3

Point Q divides the line segment AB. = 2 : 2

                                                              = 1 : 1

 Point R divides the line segment AB. = 3 : 1

Que. 10. Find the area of ​​a rhombus whose vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1), in that order.

[Hint: Area of ​​rhombus = 1/2 (product of its diagonals)]

Distance formula,