Show that the perpendicular drawn from the point `(4,1)` on the line segment joining `(6,5)a n d(2,-1)` divides it internally in the ratio `8: 5.`

Explanation

Let the perpendicular drawn from `p(4,1)` on line joining `A(2, -1)` and `B(6, 5)` divide in the ratio `k : 1` at the point R.

Using section formula, coordinates of R are

:`x=(6k+2)/(k+1) and y=(5k−1)/(k+1) ...(i)`
PR is perpendicular to AB

(slope of PR)×(slope of AB)=−1

`=>(y−1)/(x−4)×(5−(−1))/(6−2)=−1`

`=>[{(5k−1)/(k+1)}−1]/[{(6k+2)/(k+1)}−4]×6/4=−1`

`=>(5k−1−k−1)/(6k+2−4k−4)=−4/6`

`=>( 4k−2)/(2k−2)=−2/3`

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