Question

2. Find the equation of a straight line through the point of intersection of the lines 8x + 3y = 18,
4x + 5y = 9 and bisecting the line segment joining the points (5,-4) and (-7,6).​

Answer 1

Solution :

Step 1.

( Finding the point of intersection of the given lines )

The given lines are:

8x + 3y = 18 .....(1)

4x + 5y = 9 .....(2)

Multiplying (2) no. equation by 2 and subtracting from (1) no. equation, we get

8x + 3y - 8x - 10y = 18 - 18

⇒ - 7y = 0

⇒ y = 0

Putting y = 0 in (1) no. equation, we get

8x + 0 = 18

⇒ 8x = 18

⇒ x = 9/4

Thus, (9/4, 0) is the point of intersection.

Step 2.

( Finding the mid-point of the line segment joining the points (5, 4) and (- 7, 6) )

The two points are (5, 4) and (- 7, 6)

Thus, the middle point is:

((5 - 7)/2, (4 + 6)/2)

i.e., (- 2/2, 10/2)

i.e., (- 1, 5)

Step 3.

( Finding the required line joining the points mentioned )

We have to find the straight line joining the points (9/4, 0) and (- 1, 5)

∴ the required line is

(x - 9/4)/(9/4 + 1) = (y - 0)/(0 - 5)

⇒ (4x - 9)/(9 + 4) = - y/5

⇒ (4x - 9)/13 = - y/5

⇒ 5 (4x - 9) + 13y = 0

⇒ 20x - 45 + 13y = 0

⇒ 20x + 13y = 45 (Ans.)