Problems on Condition of Perpendicularity

Here we will solve various types of problems on condition of perpendicularity of two lines. 

1. Prove that the lines 5x + 4y = 9 and 4x – 5y – 1 = 0 are perpendicular to each other.

Solution:

Equation of the 1st line 5x + 4y = 9.

Now we need to express the above equation in the form y = mx + c.

5x + 4y = 9

4y = -5x + 9

y = -$\frac{5}{4}$x + $\frac{9}{4}$

Therefore, the slope (m $_{1}$) of the 1st line = -5/4

Equation of the second line 4x - 5y - 1 = 0

Now we need to express the above equation in the form y = mx + c.

4x – 5y – 1 = 0

⟹ -5y = -4x + 1

⟹ y = $\frac{4}{5}$– $\frac{1}{5}$ 

Therefore, the slope (m $_{2}$) of the 2nd line = $\frac{4}{5}$

Now,

m $_{1}$ × m $_{2}$ = $\frac{-5}{4}$  ×  $\frac{4}{5}$= -1

Therefore, the given lines are perpendicular to each other.

2. Find the value of k if the lines 7y = kx + 4 and x + 2y = 3 are perpendicular.

Solution:

The slope of the lines can be found by comparing the equations with y = mx + c.

Equation of the first straight line 7y = kx + 4

Now we need to express the given equation in the form y = mx + c.

7y = kx + 4

⟹ y = $\frac{k}{7}$x + $\frac{4}{7}$

Therefore, the slope (m$_{1}$) of the given line = $\frac{k}{7}$

Equation of the second line x + 2y = 3

Now we need to express the given equation in the form y = mx + c.

x + 2y = 3

⟹ 2y = -x + 3

⟹ y = -$\frac{1}{2}$x + $\frac{3}{2}$

Therefore, the slope (m$_{2}$) of the given line = -$\frac{1}{2}$

Now according o the problem the two given lines are perpendicular.

i.e., m$_{1}$ × m$_{2}$ = -1

⟹ $\frac{k}{7}$ × -$\frac{1}{2}$ = -1

⟹ -$\frac{k}{14}$ = -1

⟹ k = 14

Therefore, the value of k = 14

● Equation of a Straight Line

• Inclination of a Line
• Slope of a Line
• Intercepts Made by a Straight Line on Axes
• Slope of the Line Joining Two Points
• Equation of a Straight Line
• Point-slope Form of a Line
• Two-point Form of a Line
• Equally Inclined Lines
• Slope and Y-intercept of a Line
• Condition of Perpendicularity of Two Straight Lines
• Condition of parallelism
• Problems on Condition of Perpendicularity
• Worksheet on Slope and Intercepts
• Worksheet on Slope Intercept Form
• Worksheet on Two-point Form
• Worksheet on Point-slope Form
• Worksheet on Collinearity of 3 Points
• Worksheet on Equation of a Straight Line

10th Grade Math

From Problems on Condition of Perpendicularity to HOME