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EXPLANATION.

Equation of straight line passing through (8,3).

Having intercepts whose sum is 1.

As we know that,

Formula of :

Intercept form : x/a + y/b = 1.

Using this formula in the equation, we get.

Intercepts whose sum is = 1.

⇒ a + b = 1.

⇒ b = 1 - a.

Put the values in the intercept form equation, we get.

⇒ 8/a + 3/(1 - a) = 1.

⇒ 8(1 - a) + 3(a) = a(1 - a).

⇒ 8 - 8a + 3a = a - a².

⇒ 8 - 5a = a - a².

⇒ 8 - 5a - a + a² = 0.

⇒ 8 - 6a + a² = 0.

⇒ a² - 6a + 8 = 0.

Factorizes the equation into middle term splits, we get.

⇒ a² - 4a - 2a + 8 = 0.

⇒ a(a - 4) - 2(a - 4) = 0.

⇒ (a - 2)(a - 4) = 0.

⇒ a = 2  and  a = 4.

When a = 2, we get.

⇒ b = 1 - a.

⇒ b = 1 - 2.

⇒ b = - 1.

Equation is written as,

⇒ x/a + y/b = 1.

⇒ x/2 + y/(-1) = 1.

⇒ x/2 - y = 1.

⇒ x - 2y = 2.

When a = 4, we get.

⇒ b = 1 - a.

⇒ b = 1 - 4.

⇒ b = - 3.

Equation is written as,

⇒ x/a + y/b = 1.

⇒ x/4 + y(-3) = 1.

⇒ 3x - 4y = 12.

MORE INFORMATION.

Equation of straight line parallel to axis.

(1) Equation of x - axis ⇒ y = 0.

(2) Equation of a line parallel to x - axis at a distance of b ⇒ y = b.

(3) Equation of y - axis ⇒ x = 0.

(4) Equation of a line parallel to y - axis and at a distance of a ⇒ x = a.

Given sum of intercepts = 1

When x intercepts = a then y intercepts = 1 - a

Equation of the line is

[tex] \frac{x}{a} + \frac{y}{1 - a} = 1[/tex]

The line passes through (8,3)

[tex] \frac{8}{a} + \frac{3}{1 - a} = 1 [/tex]

8(1 - a) + 3a = a (1 - a)

[tex]8 - 8a + 3a = a - {a}^{2} \\ {a}^{2} - 6a + 8 = 0 \\ [/tex]

(a - 2) (a - 4) = 0

a = 2 or 4

1. When a = 2 equation of the line is

[tex] \frac{x}{2} + \frac{y}{1 - 2} = 1 \\ \frac{x}{2} - y = 1 \\ x - 2y = 2[/tex]

2. When a = 4 equation of the line is

[tex] \frac{x}{4} + \frac{y}{1 - 4} = 1 \\ \frac{x}{4} - \frac{y}{3} = 1 \\ 3x - 4y = 12[/tex]