Let the coordinate of point P(`x_(1),y_(1)`) on the line `x+5y=13`i.e.,
`P(13-5y_(1),y_(1)`).
`:.` Distance of P from the line 12x-5y+26=0,
`2=|(21(13-5y_(1))-5y_(1)+26)/(sqrt(144+25))|`
`rArr 2 = +- (156-60y_(1)-5y_(1)+26)/(13)`
`rArr -65y_(1)=-156`[taking positive sign]
`rArr y_(1)=(156)/(65)=(12)/(5)`
`,' x_(1)=13-5y_(1)`
`=13-12=1`
So, the coordinate of is `P(1,(12)/(5))`.
Similarly, the coordinates of Q are `(-3,(16)/(5))`[taking negaive sign]
(ii) Let coordinates of the point on the line x+y=4 be `(4-y_(1),y_(1))`.
Distance from the line `4x+3y-10=0`
`1=|(4(-4y_(1))+3y_(1)-10)/(sqrt(16+9))|`
`rArr 1=+-(16-4y_(1)+3y_(1)-10)/(5)` [taking negative sign]
`rArr 5=6-y_(1)`
`rArr y_(1)=1`
If `y_(1)=1`, then `x_(1)=3`
So, the point is (3,1)
Similarly, taking negative sign the point is (-7,11)
(iii) Given point A(-2,5) and B(3,1)
Now, the point P divides line joining the point A and B in 1:2.
`:' x_(1)=(1.3+2(-2))/(1+2)=(3-4)/(3)=(-1)/(3)`
and `y_(1)=(1.1+2.5)/(1+2)=(11)/(3)`
So, the coordinates of P are `(-(1)/(3),(11)/(3))`.
Thus, the point Q divided the line joining A to B in 2:1.
`because x_(2)=(2.3+1(-2))/(2+1)=(4)/(3)`
and `y_(2)=(2.1+1.5)/(2+1)=(7)/(3)`
Hence, the coordinates of Q are `((4)/(3),(7)/(3))`.
Hence, the correct matches are (i) `to` (c ), (ii) `to` (a), (iii) `to` (b).
