In what ratio is the line segment joining the points A (- 4 2 and B 8 3 divided by the Y axis also find the point of intersection?

(i) Let the required ratio be m1 : m2
Consider A(-4, 6) = (x1, y1); B(8, −3) = (x2 , y2) and letP(x, y) be the point of intersection of the line segmentAnd the y-axis

By section formula, we have,

`x=(m_1x_2+m_2x_1)/(m_1+m_2), y=(m_1y_2+m_2y_1)/(m_1+m_2)`

`=> x=(8m_1-4m_2)/(m_1+m_2), y=(-3m_1+6m_2)/(m_1+m_2)`

The equation of the y-axis is x = 0

`=> x=(8m_1-4m_2)/(m_1+m_2)=0`

`=> 8m_1-4m_2=0`

`=>8m_1=4m_2`

`=>m_1/m_2=4/8`

`=>m_1/m_2=1/2`

(ii) from the previous subpart, we have,

`m_1/m_2=1/2`

`=>m_1=k and m_2=2k, where k`

Is any constant.
Also, we have,

`=> x=(8m_1-4m_2)/(m_1+m_2), y=(-3m_1+6m_2)/(m_1+m_2)`

`=> x=(8k-4xx2k)/(k+2k), y=(-3k+6xx 2k)/(k+2k)`

`=> x=(8k-8k)/(3k), y=(-3k+12k)/(3k)`

`=>x=0/(3k), y=(9k)/(3k)`

`=>x=0, y=3`

Thus, the point of intersection is p (0, 3)(iii) The length of AB = distance between two points A and B.The distance between two given points

`A(x_1 , y_1)` and B `(x_2 , y_2)` is given by,

Distance AB `=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`=sqrt((8+4)^2+(-3-6)^2)`

`=sqrt(12^2+9^2)`

`=sqrt(144+81)`

`=sqrt(225)`

`=15` units

Text Solution

Answer : (1:2), `P(0,(7)/(3))`

Solution : Let the ratio be k:1 and let P(0,y) be the point of division.