Question

Lines 2x − by + 7 = 0 and ax − 2y − 7 = 0 are perpendicular to each other.

Statement 1: `2/a = b/2`

Statement 2: Slope of line 2x − by + 7 = 0 is `2/b`​ and slope of line ax − 2y − 7 = 0 is `a/2`.

Options

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true, and statement 2 is false.

• Statement 1 is false, and statement 2 is true.

Answer 1

Statement 1 is false, and statement 2 is true.

Explanation:

Given equation of line 2x − by + 7 = 0 and ax − 2y − 7 = 0

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

⇒ 2x − by + 7 = 0

⇒ by = 2x + 7

⇒ y = `2/bx + 7/b`

∴ Slope of the line (m₁) = `2/b`

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

⇒ ax − 2y − 7 = 0

⇒ 2y = ax − 7

⇒ y = `a/2x − 7/2`

∴ Slope of the line (m₂) = `a/2`

So, statement 2 is true.

We know that,

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

⇒ `m_1 ​× m_2​ = −1`

⇒ `2/b​ × a/2 ​= −1`

⇒ `2/b ​= ​−1 xx 2/a`

⇒ `2/b = -2/a`

So, statement 1 is false.