Question

A line 2x + 8y = 15.

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

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

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

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

Explanation:

Given, 2x + 8y = 15

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

⇒ 8y = −2x + 15

⇒ y = `−2/8x+15/8`

⇒ y = `−1/4x+15/8`

∴ Slope of the line (m) = `−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) = `−1/4​ (x - 0)`

⇒ y = `−1/4x`

⇒ 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 = `−a/bx − c/b`

∴ Slope of the line (m) = `−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) = `−a/b (x - 0)`

⇒ y = `−a/b​ x`

⇒ by = −ax

⇒ ax + by = 0

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