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Question

The perpendicular distance of a line from the origin is 5 units and its slope is − 1. Find the equation of the line.

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Solution

Let c be the intercept on the y-axis.
Then, the equation of the line is

y=-x+c m=-1x+y=cx12+12+y12+12=c12+12 Dividing both sides by coefficient of x2+coefficient of y2x2+y2=c2

This is the normal form of the given line.
Therefore, c2 denotes the length of the perpendicular from the origin.
But, the length of the perpendicular is 5 units.

c2=5c=±52

Thus, substituting c=±52 in y=-x+c, we get the equation of line to be y=-x+52 or, x+y-52=0

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