Find the equation of the lines which passes through the point (3, 4) and cuts off intercepts from the coordinate axes such that their sum is 14.

The equation of line in intercept form is:

…(i)

where a and b are the intercepts on the axis.

Given that: a + b = 14

⇒ b = 14 – a

So, equation of line is

⇒ 14x – ax + ay = 14a – a² …(ii)

If eq. (ii) passes through the point (3,4) then

14(3) – a(3) + a(4) = 14a – a²

⇒ 42 – 3a + 4a – 14a + a² = 0

⇒ a² – 13a + 42 = 0

⇒ a² – 7a – 6a + 42 = 0

⇒ a(a – 7) – 6(a – 7) = 0

⇒ (a – 6)(a – 7) = 0

⇒ a – 6 = 0 or a – 7 = 0

⇒ a = 6 or a = 7

If a = 6, then

6 + b = 14

⇒b = 14 – 6 = 8

If a = 7, then

7 + b = 14

⇒b = 14 – 7

= 7

If a = 6 and b = 8, then equation of line is

⇒ 4x + 3y = 24

If a = 7 and b = 7, then equation of line is

⇒ x + y = 7