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Question

Find all points on the line x+y=4 that lie at a unit distance from the line 4x+3y=10

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Solution

To take any point on the line x+y=4 ......(1)
Let x=α, then from (1),y=4α
P(α,4α) is any point on the line (1)
It will be a required point if its perpendicular distance from the line
4x+3y10=0 is 1 unit
|4α+3×(4α)10|42+32=1
|4α+123α10|16+9=1
|α+2|25=1
α+2=±5
α=2±5
α=3,7
Hence, the required points are (3,43) and (7,4+7) or (3,1) and (7,11)

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