Any point on the 3x−y=0 is of the form (a,3a).
∴ Distance between (a,3a) and (4,1) is 112√2.
√(a−4)2+(3a−1)2=112√2
(a2−8a+16+9a2−6a+1)=1218
10a2−14a+17=1218
10a2−14a+158=0
80a2−112a+15=0
(4a−5)(20a−3)=0
a=54,320
∴ Equation of line passing through (4,1) and (54,154) is
(y−y1)=(y2−y1x2−x1)(x−x1)
(y−1)=⎛⎜
⎜
⎜⎝154−15−4⎞⎟
⎟
⎟⎠(x−4)
(y−1)=114(x−4)
4y−4=11x−44
11x−4y=40
∴ Equation of line passing through (4,1) and (320,920) is
(y−y1)=(y2−y1x2−x1)(x−x1)
(y−1)=⎛⎜
⎜
⎜⎝920−1320−4⎞⎟
⎟
⎟⎠(x−4)
(y−1)=1177(x−4)
77y−77=11x−44
11x−77y+33=0
x−7y+3=0