Find the ratio in which the point P`((3)/(4),(3)/(12))` divides the line segment joining the points A `((1)/(2),(3)/(2))` and B `(2,-5)`.

Explanation

Let P `((1)/(2),(5)/(12))` divide AB internally in the ration M : n.
Using the section formula , we get`((3)/(4),(5)/(12))=((2m-(n)/(2))/(m+n),(-5m+(3)/(2))/(m+n))`
[`:'` internal section formula , the coordinates of point P divides the line segment joining the point `(x_(1),y_(1))and(x_(2),y_(2))` in the ratio `m_(1) :m_(2)` internally is `((m_(2)x_(1)+m_(1)x_(2))/(m_(1)+m_(2)),(m_(2)y_(1)+m_(1)y_(2))/(m_(1)_m_(2)))]`
On equating , we get `(3)/(4)=(2m-(n)/(2))/(m+n) and (5)/(12)=(-5m+(3)/(2))/(m+n)`
`rArr(3)/(4)=(4m-n)/(2(m+n))and(5)/(12)=(-10m+3n)/(2(m+n))`
`rArr(3)/(2)=(4m-n)/(m+n)and (5)/(6)=(-10m+3n)/(m+n)`
`rArr3m+3n=8n-2nand5m+5n=-60m+18n`
`rArr5n-5n=0and65-13n=0`
`rArrn=m and13(5m-n)=0`
`rArrn=m and5m-n=0`
Since, `m=n` does not satisfy.
`:. 5m-n=0`
`rArr5m=n`
`:. (m)/(n)=(1)/(5)`
Hence , the required ratio is `1 : 5`.