# Coordinate Geometry

## Exercise 7.2 Part 1

Question 1: Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2 : 3.

Solution: A = (-1, 7) and B = (4, -3)

So, x_1 = -1,  y_1 = 7,  x_2 = 4 and y_2 = -3

We also have; m_1 = 2 and m_2 = 3

Let us assume P is the point of division

Coordinates of P can be calculated as follows by using section formula:

x=(m_1x_2+m_2x_1)/(m_1+m_2)

=(2xx4+3xx-1)/(5)

=(8-3)/(5)=5/5=1

y=(m_1y_2+m_2y_1)/(m_1+m_2)

=(2xx-3+3xx7)/(5)

=(-6+21)/(5)=15/5=3

Hence, P = (1, 3)

Question 2: Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

Solution: We have A = (4, -1) and B = (-2, -3)

Here; x_1 = 4, y_1 = -1, x_2 = -2 and y_2 = -3

Let us assume that points C and D divide line segment AB into three equal parts so that AC = CD = DB

In that case, point C divides AB in ratio 1 : 2 and point D divides AB in ratio 2 : 1

For point C; m_1 = 1 and m_2 = 2

For point D; m_1 = 2 and m_2 = 1

Coordinates of point C can be calculated as follows:

x=(m_1x_2+m_2x_1)/(m_1+m_2)

=(1xx-2+2xx4)/(3)

=(-2+8)/(3)=6/3=2

y=(m_1y_2+m_2y_1)/(m_1+m_2)

=(1xx-3+2xx-1)/(3)

=(-3-2)/(3)=-5/3

Coordinates of point D can be calculated as follows:

x=(m_1x_2+m_2x_1)/(m_1+m_2)

=(2xx-2+1xx4)/(3)

=(-4+4)/(3)=0

y=(m_1y_2+m_2y_1)/(m_1+m_2)

(2xx-3+1xx-1)/(3)

=(-6-1)/(3)=-7/3

Hence; C = (2,  -5/3) and D = (0,  -7/3)

Question 3: To conduct sports day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in figure. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag? Solution: Coordinates for green flag A = (2, 25)

Coordinates for red flag B = (8, 20)

Using distance formula, length of AB can be calculated as follows:

AB=sqrt((x_2-x_2)^2+(y_2-y_1)^2)

=sqrt((8-2)^2+(20-25)^2)

=sqrt(6^2+(-5)^2))

=sqrt(36+25)=sqrt61\m

Coordinates for blue flag can be calculated by using midpoint formula:

x=(x_1+x_2)/(2)

=(8+2)/(2)=5

y=(y_1+y_2)/(2)

=(25+20)/(2)=45/2

Hence; C = (5,  45/2)

Question 4: Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).

Solution: Answer: x_1 = -3, y_1 = 10, x_2 = 6, y_2 = -8, x_3 = -1 and y_3 = 6

Let us assume that point (-1, 6) divides the line segment in ratio m1 : m2

Then x coordinate for this point can be given as follows:

-1=(-3m_2+6m_1)/(m_1+m_2)

Or, -(m_1+m_2)=-3m_2+6m_1
Or, m_1+m_2=3m_2-6m_1
Or, 7m_1=2m_2
Or, m_1/m_2=2/7=2:7

The y coordinate for this point can be given as follows:

6=(10m_2-8m_1)/(m_1+m_2)

Or, 6m_1+6m_2=10m_2-8m_1
Or, 14m_1=4m_2
Or, m_1/m_2=4/14=2:7