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Question

Find all points on the line xy=2 that lie at a distance of 5 units from the line x+2y=5

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Solution

To take any point on the line xy=2 ......(1)
Let x=α, then from (1),y=α2
P(α,α2) is any point on the line (1)
It will be a required point if its perpendicular distance from the line
x+2y=5 is 5 units
|α2(α2)5|12+12=5
|α2α+4|2=5
|α+8|2=5
α8=±52
α=8±52
α=8+522,8522
Hence, the required points are (8+522,8+5222) and (8522,85222)
or (8+522,8+5242) and (8522,85242)
or (8+522,4+522) and (8522,4522)


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