Question

A line passes through (2, -1) and is perpendicular to the line y + 3x = 6. Its y-intercept is
1. - 3
2. \(\dfrac {5} 3\)
3. \(\dfrac {-5} 3\)
4. None of these

Answer 1

Correct Answer - Option 3 : \(\dfrac {-5} 3\)

Concept:

The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. 

The equation of the line passing through the point (x1, y1) with slope m is given by 

(y - y1) = m (x - x1)

The product of slopes of perpendicular lines is - 1.

 

Calculations:

The equation of line passing through the point (x₁, y₁) with slope m₁ is given by 

(y - y₁) = m₁ (x - x₁)

The equation of line passing through the point (2, -1) with slope m1 is given by 

(y + 1) = m1 (x - 2)              ....(1)

Given line is  y + 3x = 6 

⇒y = - 3x + 6

The slope of the line y = - 3x + 6 is m₂ = - 3.

A line passes through (2, -1) with slope  m1 and is perpendicular to the line y + 3x = 6 with slope  m₂.

⇒ m1  m₂ = -1

⇒ m1 (- 3) = -1

⇒ \(\rm m_1 = \dfrac 1 3 \)

Equation (1) becomes

⇒ (y + 1) = \(\dfrac 13\)(x - 2)

⇒ y = \(\dfrac 13\)(x - 2) - 1

⇒ y =  \(\rm (\dfrac 1 3)x - \dfrac 2 3 -1\)

⇒ y =  \(\rm (\dfrac 1 3)x + \dfrac {-5} 3\)

which is of the type y = mx + b

The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. 

Hence, A line passes through (2, -1) and is perpendicular to the line y + 3x = 6. Its y-intercept is \(\dfrac {-5} 3\)