A line 2x + 8y = 15.

Assertion (A) : The equation of line passing through origin and parallel to the given line 2x + 8y = 15 is x + 4y = 0.

Reason (R) : Equation of the line passing through the origin and parallel to ax + by + c = 0 is ax + by = 0.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

Given, 2x + 8y = 15

Converting in slope-intercept form (y = mx + c), we get :

⇒ 8y = -2x + 15

⇒ y = $-\dfrac{2}{8}x + \dfrac{15}{8}$

⇒ y = $-\dfrac{1}{4}x + \dfrac{15}{8}$

∴ Slope of the line (m) = $-\dfrac{1}{4}$

By point-slope formula,

⇒ (y - y₁) = m(x - x₁)

Equation of line parallel to the line 2x + 8y = 15 and passing through origin (0, 0) :

⇒ (y - 0) = $-\dfrac{1}{4}$ (x - 0)

⇒ y = $-\dfrac{1}{4}$ x

⇒ 4y = -x

⇒ x + 4y = 0

So, assertion (A) is true.

Given, equation of line : ax + by + c = 0

Converting in slope-intercept form (y = mx + c), we get :

⇒ by = -ax - c

⇒ y = $-\dfrac{a}{b}x - \dfrac{c}{b}$

∴ Slope of the line (m) = $-\dfrac{a}{b}$

By point-slope formula,

⇒ (y - y₁) = m(x - x₁)

Equation of line parallel to ax + by + c = 0 and passing through origin (0, 0) :

⇒ (y - 0) = $-\dfrac{a}{b}$ (x - 0)

⇒ y = $-\dfrac{a}{b}$ x

⇒ by = -ax

⇒ ax + by = 0

So, reason (R) is true, and it is the correct reason for assertion (A).

Hence, option 3 is the correct option.