A normal to a line is a line segment drawn from a point perpendicular to the given line.
Let p be the length of the normal drawn from the origin to a line, which subtends an angle ø with the positive direction of x-axis as follows.
Then, we have Cos ø = p/m à m = p/Cos ø
And Sin ø = p/n à n = p/Sin ø
The equation of line in intercept form is,
x/m + y/n =1
x/(p/Cos ø) + y/(p/Sin ø) = 1
x Cos ø/p + y Sin ø/p =1
x Cos ø + y Sin ø = p.
This is called the normal form of equation of the given line making the angle ø with the positive direction of x-axis and whose perpendicular distance from the origin is p.
Converting the general equation of a line into normal form:
The equation of a straight line in general form is,
From the above equations, we have
Cos ø = p/m and Sin ø = p/n
From Trigonometric identity, we have
Cos2 ø + Sin2ø=1
p2/m2 + p2/n2=1
Solving this equation gives us,
P = mn/√(m2+n2), which is the perpendicular distance from the origin to the line x/m + y/n =1 (i.e., nx + my – mn=0)
Thus, we have the distance from origin to the line ax+by+c=0, = |c|/√(a2+b2)
Thus, for converting the given line into normal form, divide the equation ax+by+c=0 by √(a2+b2).
[a/ √(a2+b2)] x + [b/√(a2+b2)] + c/√(a2+b2)=0
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