Question

Two straight lines 3x − 2y = 15 and 2x+ ky + 8 = 0.

Assertion (A): The given two lines are perpendicular to each other and k = 3.

Reason (R): If the inclinations of two lines are α and β; then tan α = − cot β.

पर्याय

• 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 А.

• 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 incorrect reason for A.

Explanation:

Given, two lines: 3x − 2y = 15 and 2x + ky + 8 = 0

Converting the equation of first line in slope-intercept form (y = mx + c), we get:

⇒ 3x − 2y = 15

⇒ −2y = −3x + 15

⇒ y = `3/2x − 15/2`

∴ Slope of the line (m₁) = `3/2`

Converting the equation of second line in slope-intercept form (y = mx + c), we get:

⇒ 2x + ky + 8 = 0

⇒ ky = −2x − 8

⇒ y = `−2/kx − 8/k`

∴ Slope of the line (m₂) = `−2/k`

We know that,

The two lines are perpendicular if product of their slopes is −1.

⇒ `m_1 × m_2 = −1`

⇒ `3/2 × (−2/k) = −1`

⇒ `−3/k = −1`

⇒ −3 = −k

⇒ k = 3

So, assertion (A) is true.

If the inclination of two lines are α and β, then Slope of first line (m₁) = tan α and Slope of second line (m₂) = tan β.

⇒ m₁ × m₂ = −1

⇒ tan α × tan β = −1

⇒ tan α = `−1/tan β`

⇒ tan α = −cot β

So, reason (R) is true, but it is not the correct reason for assertion (A).