Straight Line

Straight Line Important Formulae
Slope: m=y2−y1x2−x1 Slope-Intercept form: y=mx+b Slope Point Format: y−y1=m(x−x1) Angle between two lines: α=tan−1⁡(m2−m11+m1m2) d=(x2−x1)2+(y2−y1)2+(z2−z1)2
After successful completion of this topic, you should be able to:

  • Find and interpret equation of a straight line in various forms.
  • Perform slope calculations including parallel and perpendicular lines.
  • Find angle between two coplanar, non-parallel lines.
  • Calculate distance between two points in space.

Slope of a Straight Line

Slope of a straight line gives you an idea about its inclination with reference to x-axis. Slope is also referred as gradient.

Slope of a straight line=RiseRun=ΔyΔx

Equation of a Straight Line

The equation of a straight line (or any curve) is the relation between the x and y (and z) coordinates of all points lying on it.
The general form of the equation of a straight line is:


Various forms of equations of Straight Line

Slope Point Format:



where x1 and y1 are the coordinates of the point through which the line passes.

Slope Intercept Format:


where m = slope and b = y-intercept
For the above line, y-intercept = 1, and
slope = RiseRun = 24 = 12
So the equation will be, y=0.5x+1 2y=x+2 x−2y+2=0

Double Intercept Format:


xa+yb=1

where a = x intercept and b = y intercept

Parallel and Perpendicular Lines

For parallel lines slopes are equal i.e. m1=m2
For perpendicular lines
m1∗m2=−1 or,


Slopes of perpendicular lines
Brigban, CC0, via Wikimedia Commons

Angle between Two Lines

Angle between two straight lines is given by:


Distance Formula

Distance between two points in the space P1 (x1,y1,z1) and P2 (x2,y2,z2) is given by:

This can be proved by repeated application of the Pythagorean Theorem.

Distance Formula
Jim.belk, Public domain, via Wikimedia Commons

The distance between the points (0,0) and (a,b) is = b. This means:



Correct Answer: B

Given the slope-intercept form of a line as y = mx + b, which one of the following is true?



Solution:
For the straight line y = mx + b, m is the slope and b is the y-intercept.
To get x-intercept,
We will have to substitute y=0: y=mx+b0=mx+b−bm=x

Correct Answer: D

The maximum degree (power) of x and y terms in an equation of straight line is:



Solution:

In an equation of a straight line, the maximum power of x and y terms is one. In an equation of a conic section (ellipse, parabola and hyperbola), the maximum power of x and y terms is two.

Correct Answer: B

Two non-vertical lines are parallel if and only if they have the same:



Solution:

Parallel lines have same slope. (m1 = m2)

Perpendicular lines have product of their slopes = -1

Correct Answer: A

The slope of a straight line passing through origiin and a point (x1, y1) is given by:



Solution:
Since the line is passing through Origin (0,0). The slope will be: Slope=y2−y1x2−x1=0−y10−x1=y1x1

Correct Answer: D

The distance formula can be extended to find the distance between two points in:



Correct Answer: C

The distance between the points (0,0) and (a,b) is = b. This means:



Correct Answer: B

Given the slope-intercept form of a line as y = mx + b, which one of the following is true?



Solution:
For the straight line y = mx + b, m is the slope and b is the y-intercept.
To get x-intercept,
We will have to substitute y=0: y=mx+b0=mx+b−bm=x

Correct Answer: D

If the distance between the points (3, 4) and (a, 2) is 8 units then find the value of a:



Solution:
The distance between the points (3, 4) and (a, 2) is 8. (3−a)2+(4−2)2=8(3−a)2+(4−2)2=82(3−a)2+4=64(3−a)2=60(3−a)=±2153±215=a

Correct Answer: A

The angle between lines 2x- 9y + 16=0 and x + 4y + 5 = 0 is given by:



Solution:
First let's find out slopes of each line. 2x−9y+16=0y=29x+169 which means slope m1 =29 Similarly, slope of the second line, m2 = −14 The angle between two lines is given by: α=tan−1⁡(m2−m11+m1m2)=tan−1⁡(−14−291+29×−14)=tan−1⁡(−17363436)=tan−1⁡(−12)

Correct Answer: C

What is the equation of the straight line which passes through the point (1, 2) and cuts off equal intercepts from the axes ?



Solution:
Method I: Use double intercept form xa+yb=1 But both intercepts are equal. xa+ya=1 But the line passes through (1,2) 1a+2a=1 1+2=a 3=a Substituting, x3+y3=1 x+y=3 Method II: Since it cuts equal intercepts, the slope will be m = -1.

It passes through (1,2). y−y1=m(x−x1)y−2=(−1)(x−1)y−2=−x+1x+y−3=0x+y=3

Correct Answer: A

What is the equation of the straight line which is perpendicular to y = x and passes through (3, 2)?



Solution:
Slope of the given line y = x: m1 = 1
Slope of the perpendicular line: m2 = −1m1 = -1
y−y1=m(x−x1) But it passes through (3,2) y−2=(−1)(x−3)y−2=−x+3)x+y=5

Correct Answer: B

If the co-ordinates of the middle point of the portion of a line intercepted between coordinate axes (3,2), then the equation of the line will be:





Solution:

As shown in the figure, the coordinates of intersection point of the straight line with x- axis will be:

B (2×3, 0)= B (6,0)

and y-axis will be:

A(0, 2×2) = A(0,4).

So, x-intercept = a = 6, y-intercept = b= 4
Using double -intercept form of straight line equation, xa+yb=1x6+y4=12x+3y=12

Correct Answer: A

The area of triangle formed by the lines x=0,y=0 and xa+yb=1, is:



Solution:

The equation of the line xa+yb=1 is in double intercept form. This line makes an intercept 'a' on the x-axis and 'b' on the y-axis.

Area of the right angled triangle is = 12 × (base) × (Height) = 12 × a × b.

Correct Answer: B

The slope of a straight line which makes 30∘ angle with x-axis is:



Solution:
Slope=tan⁡θ=tan⁡30∘=13

Correct Answer: B

The distance between the points (-5,-5,7) and (3,0,5) is:



Solution:

d=(3−(−5))2+(0−(−5))2+(5−7)2=93=9.643

Correct Answer: C

If the distance between the points (3, 4) and (a, 2) is 8 units then find the value of a:



Solution:
The distance between the points (3, 4) and (a, 2) is 8. (3−a)2+(4−2)2=8(3−a)2+(4−2)2=82(3−a)2+4=64(3−a)2=60(3−a)=±2153±215=a

Correct Answer: A

The angle between lines 2x- 9y + 16=0 and x + 4y + 5 = 0 is given by:



Solution:
First let's find out slopes of each line. 2x−9y+16=0y=29x+169 which means slope m1 =29 Similarly, slope of the second line, m2 = −14 The angle between two lines is given by: α=tan−1⁡(m2−m11+m1m2)=tan−1⁡(−14−291+29×−14)=tan−1⁡(−17363436)=tan−1⁡(−12)

Correct Answer: C