Question

Find the equation of the straight line perpendicular to the line x + 2y = 4, which cuts an intercept of 2 units from the positive y-axis. Hence, find the intersection point of the two lines.

Answer 1

Given: The line x + 2y = 4 and the required line is perpendicular to it and cuts an intercept 2 units from the positive y-axis (y-intercept = 2).

Step-wise calculation:

1. Rewrite x + 2y = 4 as `y = -(1/2)x + 2`.

So, its slope `m_1 = -1/2`.

2. Slope of a line perpendicular to it is `m_2 = -1/(m_1) = 2`.

3. The required line has slope 2 and y-intercept 2.

So, its equation is y = 2x + 2 or 2x – y + 2 = 0.

4. Find the intersection with x + 2y = 4 by substituting y = 2x + 2:

x + 2(2x + 2) = 4

⇒ x + 4x + 4 = 4

⇒ 5x = 0

⇒ x = 0

Then y = 2(0) + 2 = 2.

Equation of the required line: y = 2x + 2.

Intersection point of the two lines: (0, 2).