Question

22. The ratio in which (4, 5) divides the join of (2, 3)

and (7, 8) is :
(a) 4 : 3
(c) 3 : 2
(b) 5:2
(d) 2:3
​

Answer 1

Let the ratio be m:n

• (x,y)=(4,5)
• Points be (x1,y1)=(2,3)
• (x2,y2)=(7,8)

We know

`\boxed{\sf (x,y)=\left((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n)\right)}`

`\\ \sf\longmapsto (4,5)=\left((7m+2n)/(m+n),(8m+3n)/(m+n)\right)`

Now

.
`\\ \sf\longmapsto (7m+2n)/(m+n)=4\dots(1)`

`\\ \sf\longmapsto (8m+3n)/(m+n)=5\dots(2)`

Adding both

`\\ \sf\longmapsto (7m+2n+8m+3n)/(m+n)=4+5`

`\\ \sf\longmapsto (7m+8m+2n+3n)/(m+n)=9`

`\\ \sf\longmapsto (15m+5n)/(m+n)=9`

`\\ \sf\longmapsto 15m+5n=9(m+n)`

`\\ \sf\longmapsto 15m+5n=9m+9n`

`\\ \sf\longmapsto 15m-9m=9n-5n`

`\\ \sf\longmapsto 6m=4n`

`\\ \sf\longmapsto (m)/(n)=(6)/(4)`

`\\ \sf\longmapsto (m)/(n)=(3)/(2)`

`\\ \sf\longmapsto m:n=3:2`

Option B is correct