Math Doubts

Equation of a straight line in terms of X-intercept and Y-intercept

Expressing equation of a straight line in terms of X-intercept and Y-intercept when the straight line intersects both axis is defined Equation of a straight line in terms of X-intercept and Y-intercept.

It is another special case. You have seen straight line with X-intercept and slope and also seen straight line with Y-intercept and slope but this case this entirely different to them. Straight lines may also pass through both axis at an X-intercept as well as at a Y-intercept. It can be also be denoted by the General form of the straight line but it is transformed into X-intercept and Y-intercept form when both intercepts are substituted in the standard equation of the straight line.

Geometrical Explanation

Assume,

AB

is a straight line and it crosses the horizontal x-axis at an intercept and also it passes through the vertical y-axis at another intercept.

Straight line passes through x-axis and y-axis

The straight line

AB

is intersected the horizontal x-axis at a point

A

at an x-intercept of

a

units. Therefore, the point

A

is located at

(a, 0)

in the Cartesian coordinate system. Similarly, the straight

AB

is intersected the vertical y-axis at a point

B

at a y-intercept of

b

units. So, the point

B

is located at

(0, b)

in geometric coordinate system. Thus, the straight line

AB

is formed a right angled triangle

ΔBAO

. Assume, the straight line

AB

is making an angle theta

(θ)

with the horizontal x-axis.

Straight line passes through x-axis and y-axis with some angle

According to right angled triangle

ΔBAO

.

The line segment

AB

is known hypotenuse of the right angled triangle

ΔBAO

.

The line segment

OA

is known adjacent side of the right angled triangle

ΔBAO

.

The line segment

OB

is known opposite side of the right angle triangle

ΔBAO

.

Assume, the angle between hypotenuse

AB

and adjacent side

OA

is alpha

α

. The angle of the right angled triangle

ΔBAO

is

BAO = α

.

Consider a point on the straight line

AB

and assume it to call point

C

and also the coordinates of the point

C

is

(x, y)

. The point

C(x, y)

represents each and every point on the line including the points

A

and

B

. Now, draw a horizontally parallel line from point

C

towards vertical axis. Assume, the parallel line perpendicularly meet the y-axis at a point, assumed to call point

D

. Thus, another right angled triangle

ΔBCD

is formed by the part of the straight line

AB

.

Straight line passes through x-axis and y-axis with some angle forms similar triangles

Assume, point

E

is a point on the horizontal x-axis. The exterior angle of the right angled triangle

ΔBAO

is

BAE = θ

. The summation of the angle of the triangle and exterior angle is

180°

because the summation of the angles forms a straight angle.

Therefore,

θ+α = 180°

α = 180° θ

According to right angled triangle

ΔBAO

,

tanα =   OBOA

The length of the opposite side

(OB)

of the right angled triangle

ΔBAO

is

OB = b

.

The length of the adjacent side

(OA)

of the right angled triangle

ΔBAO

is

OA = a

.

tanα =   OBOA = ba

Now replace the angle alpha

(α)

in terms of theta

(θ)

.

tan(180° θ) =   ba

180° θ

means, tangent function is brought to second quadrant. In second quadrant, the trigonometric ratio tangent is negative.

tan θ = ba

tan θ = ba

According to the concept of slope of a straight line, slope of a straight line

(m) = tan θ

.

m = ba

Similarly, as per the right angled triangle

ΔBCD

,

tanα =   DBDC

The length of the opposite side

(DB)

of the right angled triangle

ΔBCD

is

DB = OBOD = by

.

The length of the adjacent side

(DC)

of the right angled triangle

ΔBCD

is

DC = x

.

tanα =   DBDC = byx

tan(180° θ) =   DBDC = byx

tanθ = byx

tanθ = (byx)

m = (byx)

Finally, the slope of the same straight line

AB

is obtained in terms of the coordinates of the points of the line and slope of the straight line from both triangles

ΔBAO

and

ΔBCD

.

m = ba

and

m = (byx)

. These two expressions are equal in value because they both represent slope of the same straight line.

ba = (byx)

ba = byx

xa = byb

xa = bb yb

xa + yb = bb

xa + yb = 1

This algebraic expression represents a straight line when the straight passes through both axis of the Cartesian coordinate system through at x-intercept and y-intercept.



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more