The formula used to find the length of perpendicular from origin to the line x/a + y/b = 1 is:
$\frac{|(0/a) + (0/b) - 1|}{\sqrt{(1/a)^2 + (1/b)^2}} = \frac{\sqrt{(b^2 + a^2)}}{\sqrt{(a^2 + b^2)}} = 1$
Which formula is used to find the length of perpendicular from origin to the line x/a + y/b = 1?
3 answers
The line x/a + y/b = 1 can be written as
bx + ay = ab
This line has slope -b/a, so the perpendicular line has equation
y = a/b x
The lines intersect at
x = ab^2/(a^2+b^2)
y = a^2b/(a^2+b^2)
so the distance from (0,0) to the intersection is
d = |ab|/√(a^2+b^2)
bx + ay = ab
This line has slope -b/a, so the perpendicular line has equation
y = a/b x
The lines intersect at
x = ab^2/(a^2+b^2)
y = a^2b/(a^2+b^2)
so the distance from (0,0) to the intersection is
d = |ab|/√(a^2+b^2)
Therefore, the length of the perpendicular from origin to the line x/a + y/b = 1 is:
|a/b * x - y| / sqrt(a^2 + b^2) = |a/b * (ab^2/(a^2+b^2)) - (a^2b/(a^2+b^2))| / sqrt(a^2 + b^2)
Simplifying this expression, we get:
|ab| / sqrt(a^2 + b^2)
Therefore, the length of the perpendicular from origin to the line x/a + y/b = 1 is |ab| / sqrt(a^2 + b^2).
|a/b * x - y| / sqrt(a^2 + b^2) = |a/b * (ab^2/(a^2+b^2)) - (a^2b/(a^2+b^2))| / sqrt(a^2 + b^2)
Simplifying this expression, we get:
|ab| / sqrt(a^2 + b^2)
Therefore, the length of the perpendicular from origin to the line x/a + y/b = 1 is |ab| / sqrt(a^2 + b^2).