Find the distance of the point `(2,3)` from the line `2x-3y+9=0` measured along a line `x-y+1=0.`

Options

A. 4
B. `2sqrt(2)`
C. `8sqrt(2)`
D. `4sqrt(2)`

Answer

D

Explanation

The slope of the line `x-y+1=0`. So, it makes and angle of `45^(@)` with x-axis.
The equation of a line passing through (2,3) and making an angle of `45^(@)` is given by
`(x-2)/(cos 45^(@))=(y-3)/(sin 45^(@))=r`
Co-ordinate of any point on his line are ltbr? `(2+r cos 45^(@), 3+r sin 45^(@))` or, `(2+(r)/(sqrt(2)),3+(r)/(sqrt(3)))`
If this point lies on the line `3=2x-3y+9=0`
`4+rsqrt(2)-9-(3r)/(sqrt(2))+9=0 implies r=4sqrt(2)` units.