Let us the slope of the line passing through the point P(1,4) be m.

We know that equation of a straight line passing through the point (x₁,y₁) and having slope m is given by:

⇒ y - y₁ = m(x - x₁)

The equation of the straight line is:

⇒ y - 4 = m(x - 1)

⇒ mx - y = m - 4

⇒

⇒

This resembles the standard form , where a is x - intercept and b is y - intercept.

Here x - intercept and b = 4 - m

According to the problem, we need sum of intercepts to be minimum,

Let us take the sum of intercepts to be S,

⇒ S = a + b

⇒

⇒

Let us assume S is the function of m,

We know that for maxima and minima,

⇒

⇒

⇒

⇒ 4 - m² = 0 (∵ m²>0)

⇒ m = ±2

Differentiating S again

⇒

⇒

At m = - 2

⇒

⇒

⇒ >0(Minima)

At m = + 2

⇒

⇒

⇒ <0(Maxima)

We have got minima for m = - 2

Using this value we find the sum of intercepts:

⇒

⇒

⇒ S_min = 3 + 6

⇒ S_min = 9

∴ The least value of sum of intercepts is 9.