Question

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b , then show that .

Answer 1

The length of the perpendicular from the origin to the line is

p.

The intercepts of the line on x axis and y axis are a and b respectively.

The formula for the equation of the line making an intercept of a and b with x axis and y axis respectively is given by,

x a + y b =1 bx+ay ab =1 bx+ay=ab bx+ay−ab=0 (1)

The formula for the perpendicular distance of the line Ax+By+C=0 from a point ( x 1 , y 1 ) is given by,

d= | A x 1 +B y 1 +C | A 2 + B 2 (2)

Compare equation (1) with the general form of equation of line Ax+By+C=0.

A=b, B=a, C=−ab

Substitute the values of d as p, ( x 1 , y 1 ) as ( 0,0 ) and values of A, B, C from above in equation (2).

p= | b×0+a×0+( −ab ) | b 2 + a 2 p= | −ab | b 2 + a 2

Take square on both the sides.

p 2 = ( −ab ) 2 b 2 + a 2 = a 2 b 2 a 2 + b 2 1 p 2 = a 2 + b 2 a 2 b 2 = a 2 a 2 b 2 + b 2 a 2 b 2

Further simplify the above expression.

1 p 2 = 1 b 2 + 1 a 2

Thus, for a line whose intercepts on x and y axes are a and b with p being the length of perpendicular from the origin 1 p 2 = 1 b 2 + 1 a 2 .