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Question
Reduce the following equation into normal form. Find their perpendicular distances from the origin and angle between perpendicular and the positive x-axis.
x − y = 4
Solution
The given equation is x – y = 4.
It can be reduced as 1.x + (–1) y = 4
on dividing both sides by `sqrt(1^2 + (-1)^2) = sqrt2`, we obtain `1/sqrt2 "x" + (-1/sqrt2)"y" = 4/sqrt2`
⇒ `"x" cos(2π - π/4) + "y" sin(2π - π/4) = 2sqrt2`
⇒ `"x" cos 315° + "y" sin 315° = 2sqrt2` .........(i)
Eqation (i) is in the normal form.
On comparing equation (i) with the normal form of the equation of line
x cos ω + y sin ω = p, we obtain ω = 315° and p = `2sqrt2`.
Thus, the perpendicular distance of the line from the origin is `2sqrt2` while the angle between the perpendicular and the positive x-axis is 315°.
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