If `A(2,-1)a n dB(6,5)` are two points, then find the ratio in which the food of the perpendicular from `(4,1)` to `A B` divides it.

Options

A. `8:15`
B. `5:8`
C. `-5:8`
D. `-8:5`

Answer

B

Explanation

Let P (4,1) and PD ` bot `AB
Clearly , equation of AB passing through (2,-1) and (6,5) is 3x -2y -8=0
and equation of PD is 2x + 3y -11=0
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Let line AB is divided by PD in the ratio ` lamda :1 ` then intersecting point D ` ((6 lamda+2)/(lamda+1) , (5lamda -1)/(lamda +1))` lies on 2x +3y -11=0
` 2((6 lamda +2)/(lamda +1)) + 3 ((5 lamda -1)/(lamda +1)) - 11=0` ltbr. ` Rightarrow 16 lamda - 10=0 Rightarrow lamda = 5/8 `
Hence, `lamda :1 = 5 :8`