Question

If the line segment joining the points (3, −4), and (1, 2) is trisected at points P (a, −2) and Q \[\left( \frac{5}{3}, b \right)\] , Then,

 

Options

• \[a = \frac{8}{3}, b = \frac{2}{3}\]

   

• \[a = \frac{7}{3}, b = 0\]

   

• \[a = \frac{1}{3}, b = 1\]

   

• \[a = \frac{2}{3}, b = \frac{1}{3}\]

   

Answer 1

We have two points A (3,−4) and B (1, 2). There are two points P (a,−2) and Q `(5/3,b)`which trisect the line segment joining A and B.

Now according to the section formula if any point P divides a line segment joining  `A(x_1 ,y_1)" and B "(x_2 , y_2) ` in the ratio m: n internally than,

`P ( x , y ) = ((nx_1 + mx_2) /(m+n) , (ny_1 + my_2) /(m+n) )`

The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2

Now we will use section formula to find the co-ordinates of unknown point A as,`

`p( a , -2) = ((2(3) + 1 (1) )/(1+2) , (2(-4)+1(2))/(1+2))`

               `  = (7/3,-2)`

Equate the individual terms on both the sides. We get,

`a = 7/3`

Similarly, the point Q is the point of trisection of the line segment AB. So, Q divides AB in the ratio 2: 1

Now we will use section formula to find the co-ordinates of unknown point A as,

`Q (5/3 , b) = ((2(1)+1(3))/(1+2) , (2(2) + 1(-4))/(1+2))`

                `= (5/3 , 0)`

Equate the individual terms on both the sides. We get,

b = 0