Slope of a Line MCQ Quiz - Objective Question with Answer for Slope of a Line - Download Free PDF

Calculation:

Given,

Line 1:

Line 2:

Slope of Line 1,

Slope of Line 2,

For perpendicular lines, .

mn -1 = 0

Hence, the correct answer is Option 1.

Concept:

When two lines are perpendicular, then the product of their slope is -1.

i.e. m₁ × m₂ = -1

When the slope of a line is m then the slope of a line perpendicular to it = - 1/m

Calculation:

Given:

-2x + 3y + 4 = 0

y = 

The slope of the line = 

The line perpendicular to the above line will have a slope of =  = 

option (1) 3x + 2y – 4 = 0 

• The slope of the line is -3/2

option (2) 3x – 2y + 4 = 0

• The slope of the line is 3/2

option (3)  -3x + 2y + 7 = 0

• The slope of the line is 3/2

option (4) 3x – 2y – 7 = 0

• The slope of the line is 3/2

Hence option (1) 3x + 2y – 4 = 0 is correct.

Explanation:

Slope of diagonal along the line x – 2y = 1

⇒m₁ = 1/2

Slope of the diagonal along the line 4x + 2y = 3

⇒m₂ = -2

Now,  m₁m₂ = 

Then, Diagonals are perpendicular. 

∴ The quadrilateral ABCD is a rhombus.

Concept:

Angle Between Two Lines and Y-Intercept:

• The problem involves finding the y-intercept of a line L passing through a given point and making an angle of 60° with another line.
• The angle between two lines is given by the formula:
• tan(θ) = |(m₁- m₂) / (1 + m₁*m₂)|, where m₁ and m₂ are the slopes of the two lines.
• Once the equation of the line L is determined, the y-intercept can be found.

Calculation:

Given the line 2x - y + 3 = 0, we first calculate the slope of this line:

Rearranging the equation in slope-intercept form:

y = 2x + 3

The slope (m1) of this line is 2.

The line L passes through the point (2, 0) and makes an angle of 60° with the given line. The slope of line L, m2, can be calculated using the angle formula:

tan(60°) = |(m₂ - 2) / (1 + 2*m₂)|

Since tan(60°) = √3, the equation becomes:

√3 = |(m₂ - 2) / (1 + 2m₂)|

We now solve this equation for m₂:

√3(1 + 2m₂) = |m₂ - 2|

For simplicity, assume m₂ - 2 > 0 (since the angle is acute and the line makes an acute angle with the positive X-axis,   m2 > 2). Hence:

√3(1 + 2 m2) =  m2 - 2

√3 + 2√3* m2 =  m2 - 2

2√3* m2 -  m2 = -2 - √3

 m2(2√3 - 1) = -2 - √3

 m2 = (-2 - √3) / (2√3 - 1)

Now we calculate the y-intercept of the line L. The equation of the line L is:

y - 0 =  m2(x - 2)

y =  m2(x - 2)

Substitute x = 0 to find the y-intercept:

y =  m2(0 - 2)

y = -2 *  m2

Substitute the value of  m2 into this equation to calculate the y-intercept.

Hence, the y-intercept of the line L is:

The correct answer is 16 - 10√3 / 11

Concept Used:

1. A line with an undefined slope is a vertical line.

2. The slope of a line is given by m = tan θ, where θ is the angle made by the line with the positive X-axis.

3. The angle between two lines with slopes m₁ and m₂ is given by

Calculation:

Given:

Line L passes through (-2, -3) and its slope is undefined.

Since L is vertical, its equation is x = -2.

The slope of the line ax - 2y + 3 = 0 is

The angle between L and ax - 2y + 3 = 0 is 45°.

Since L is vertical, the line ax - 2y + 3 = 0 must be inclined at 45° or 135° to the y-axis.

⇒ = tan 45° = 1 or = tan 135° = -1

Since a > 0, = 1 ⇒ a = 2

The line x + ay - 4 = 0 becomes x + 2y - 4 = 0.

The slope of this line is

Let θ be the angle made by this line with the positive X-axis.

⇒ tan θ =

Since the slope is negative, the angle is obtuse.

⇒ θ =

∴ The angle made by the line x + ay - 4 = 0 with the positive X-axis is

Hence option 1 is correct.

Concept:

• The slope of a line passing through the distinct points (x₁, y₁) and (x₂, y₂) is 
• When two lines are perpendicular, the product of their slope is -1. If m is the slope of a line, then the slope of a line perpendicular to it is -1/m.

Calculation:

Let the slope of the line 2x – 5y + 4 = 0 be m₁ and the slope of the line joining the points (1, 5) and (α, 3) be m₂ 

Now, the slope of the line = m₁ = 2/5

Given lines are perpendicular to each other,

∴ m1 m2 = -1

⇒ -4 = -5 × (α -1)

⇒ (α -1) = 4/5

⇒ α = (4/5) + 1 = 9/5

Concept:

The slope of the line joining the points (x1, y1) and (x2, y2) is given by: 

Calculation:

Given: The slope of a line joining the points A (1, x) and B (3, 2) is 8

As we know, The slope of the line joining the points (x1, y1) and (x2, y2) is given by: 

⇒ 8 = 

⇒ 8 × 2 = 2 - x

⇒ 16 = 2 - x

∴ x = -14

Concept:

If θ is the inclination of the line l, then

The slope of a line is denoted by m = tan θ, θ ≠ 90°

Calculation:

Given: Line makes 30° with respect to the x-axis in a positive direction

∵ the inclination of the line is 30° i.e θ = 30° 

As we know that slope of a line is given by: m = tan θ 

So, the slope of the given line is m = tan 30° = 

Concept:

If two lines with slopes m and n are perpendicular to each other, then mn = -1.

The slope of a line passing through the points (x₁, y₁) and (x2, y2) is given by: m = .

Calculation:

The slope of the line through the points (-1, 2) and (3, 6) is 

The slope of the line through the points (4, 8) and (x, 12) is 

Using the product of slopes of perpendicular lines, we get:

⇒ 4 = (4 - x)

⇒ x = 0

Concept:

The general equation of a line is y = mx + c, where m is the slope. 

Calculation:

The given equation is 4x + 3y - 4 = 0

⇒ 3y = -4x + 4

⇒ y = (-4/3)x + 4/3

On compairing it with y = mx + c, we get

Slope, m = -4/3

Hence, the slope of the line is .

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: 

CALCULATION:

Given: The slope of a line passing through the points (2, 5) and (x, 3) is 2

Here, we have to find the value of x.

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: 

Here, x1 = 2, y1 = 5, x2 = x, y2 = 3 and m = 2.

⇒ 

⇒ 2x - 4 = - 2

⇒ 2x = 2

⇒ x = 1

Hence, option B is the correct answer.

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: 

• If two lines are parallel then their slope is same.

• If two lines L₁ and L₂ with slopes m₁ and m₂ respectively are perpendicular then m₁ × m₂ = - 1

CALCULATION :

Here, we have to find the value of x such that the lines through the points (- 2, 6) and (x, 8) is perpendicular to the line through the points (3, - 3) and (5, - 9).

Let's find out the slope of the line through the points (3, - 3) and (5, - 9).

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: 

Here, x1 = 3, y1 = - 3, x2 = 5, y2 = - 9.

⇒ 

∵ The lines through the points (- 2, 6) and (x, 8) is perpendicular to the line through the points (3, - 3) and (5, - 9)

So, let the slope of line through the points (- 2, 6) and (x, 8) be m₂

⇒ - 3 × m2 = - 1

⇒ m₂ = 1/3

So, the slope of the line through the points (- 2, 6) and (x, 8) is 1/3

⇒ 

⇒ x + 2 = 6

⇒ x = 4

Hence, option B is the correct answer.

Concept:

Collinearity of three points:

If A, B and C are any three points in the XY-Plane, then they will lie on a line, i.e., collinear if and only if slope of AB = slope of BC

The slope of the line passing through (x1, y1) and (x2, y2) is given by m = 

Calculation:

Let four points are  A(-1, -2), O(0, 0), B(1, 2) and C(2, 4)

Now,

m₁ = slope of OA = 2

m₂ = slope of OB = 2

m₃ = slope of OC = 2

Since m1 = m2 = m3 

So, the points (-1, -2), (0, 0), (1, 2) and (2, 4) are collinear

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x₁, y₁) and (x₂, y₂) is:

CALCULATION:

Here, we have to find the slope of the a line which passes through the points (- 2, 3) and (8, - 5)

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: 

Here, x₁ = - 2, y₁ = 3, x₂ = 8 and y₂ = - 5

So, the slope of the given line is 

Hence, option A is the correct answer.

CONCEPT:

The slope of a line is given by tan α and it is denoted by m.

i.e m = tan α, where α ≠ π/2 and it represents the angle which a given line makes with respect to the X – axis in the positive direction.

Note:

• Slope of X –axis or a line parallel to X – axis is : m = tan 0° = 0

• Slope of Y – axis or a line parallel to Y – axis is: m = tan π/2 = ∞

• The slope of the line joining the points (x1, y1) and (x2, y2) is: 

• If two lines are parallel then their slope is same.

CALCULATION:

Here, we have to find the value of y such that the line through the points (5, y) and (2, 3) is parallel to the line through the points (9, - 2) and (6, - 5)

Let's find the slope of the line  through the points (9, - 2) and (6, - 5)

As we know that, the slope of the line joining the points (x1, y1) and (x2, y2) is: 

Here, x1 = 9, y1 = - 2, x2 = 6, y2 = - 5.

⇒ 

∵ The line through the points (5, y) and (2, 3) is parallel to the line through the points (9, - 2) and (6, - 5)

So, the slope of the line through the points (5, y) and (2, 3) is also 1.

So, know x1 = 5, y1 = y, x2 = 2, y2 =  3 and m = 1.

⇒ 

⇒ - 3 = 3 - y

⇒ y = 6

Hence, option A is the correct answer.