A line cuts equal intercepts with both the axes and passes through the point (6, 6). The equation of the line is :

1. x - y = 12

2. x - y = 0

3. x + y = 6

4. x + y = 12

By intercept form :

Equation of line : $\dfrac{x}{a} + \dfrac{y}{b} = 1$ ………(1)

Given,

Line cuts equal intercepts with both the axes.

∴ x-intercept (a) = y-intercept (b) = p (let)

Given,

Line passes through point (6, 6). So it will satisfy the equation (1).

Substituting values in equation we get :

$\Rightarrow \dfrac{6}{p} + \dfrac{6}{p} = 1 \\[1em] \Rightarrow \dfrac{12}{p} = 1 \\[1em] \Rightarrow p = 12.$

∴ a = b = 12.

Substituting value of a and b in equation (1), we get :

$\Rightarrow \dfrac{x}{12} + \dfrac{y}{12} = 1 \\[1em] \Rightarrow \dfrac{x + y}{12} = 1 \\[1em] \Rightarrow x + y = 12.$

Hence, Option 4 is the correct option.