For what values of a and b the intercepts cut off on the coordinate axes by the line ax + by + 8 = 0

Given:

Intercepts cut off on the coordinate axes by the line ax + by +8 = 0 …… (i)

And are equal in length but opposite in sign to those cut off by the line

2x – 3y +6 = 0 ……(ii)

We know that, the slope of two lines is equal

The slope of the line (i) is \( \dfrac{-a}{b}\)

The slope of the line (ii) is \( \dfrac{2}{3}\)

So let us equate,

\( \dfrac{-a}{b}= \dfrac{2}{3}\)

a = \( \dfrac{-2b}{3}\)

The length of the perpendicular from the origin to the line (i) is

By using the formula,

The length of the perpendicular from the origin to the line (ii) is

By using the formula,

It is given that, d₁ = d₂

b = 4

So, a = \( \dfrac{-2b}{3}\)

= \( \dfrac{-8}{3}\)

The value of a is \( \dfrac{-8}{3}\) and b is 4.