Shift of Origin

Introduction 

Analytical geometry makes use of the ideas of geometry and algebra. It may include coordinate geometry, coordinate planes, equation of lines, the slope of a line, the distance between two points, distance between two planes, the shift of origin or distance, and section formulae. This chapter reviews the meaning of shift of origin, its formulae, and solved examples. Check out this article to know more about the topic!

What is shifting of origin?

As discussed above, analytical geometry is a vast subject in itself and includes many inter-related topics. Some problems may involve finding the slope of a line, shifting of origin, rotation of axes, all in a single question. However, some difficulties in coordinate geometry may be difficult to solve if we take the origin as (0,0). As a solution, the concept of shifting of origin has been introduced. It is also known as the transformation of axes in many books. The general meaning of shifting of origin is to transfer the origin to a different point without changing the orientation of the axes. When we change the coordinates of the origin, it is evident that all points on the graph will also vary concerning the new origin. 

How to form new coordinates using the shifting of origin theory?

Let us take an example to understand shifting of origin. We have a point P with the coordinates (x,y), and the origin has been shifted to (h,k). It means that the new x-axis and y-axis are now at an h and k from the original axes. So, the new length of the point P from the new X-axis will be (x – h), and the new Y-axis will be (y – k). This implies that the new coordinates of the point P will be (x – h, y – k).

To sum up, if (X, Y) are the fresh coordinates of the point with respect to the new origin (h, k), then 

X = x – h and Y = y – k 

Or,

x = X + h and y = Y + k

Shifting of origin in 3-D

Assume a point P with coordinates (x,y,z). The origin has now been shifted to (h,k,t). The new coordinates of P will be (x’, y’, z’). We can write the new coordinates as:

x ′ =x−h, y ′ =y−k and z ′ =z−t

Equation for line in new plane after shifting of origin

For a line that passes through (x1 ,y1 ,z1) and (x2 ,y2 ,z2), the equation of the line will be:

(x-x1 )/(x2 -x1 )=(y-y1 )/(y2 -y1 ).

After shifting of origin, the equation of line will now be:

[x-(x1 -h)]/(x2 -x1 )=[y-(y1 -k)]/(y2 -y1 ).

The concept of origin shifting is usually paired with the rotation of axes. After a basic understanding of these topics, the problems become more manageable. 

Solved examples 

Example 1: Find the new equation of the following curve after the coordinates are transformed: x + 3y = 6, when the origin is transferred to the unique point (–4, 1).

Solution: Using the expressions for shifting of origin, we have x = X – 4 and y = Y + 2. Using these values in the given equation, we get 

X – 4 + 3(Y + 1) = 6 ⇒ X + 3Y = 7

Hence, the new equation becomes: X+3Y=7.

Example 2: Find the new coordinates of the point (3, 4) when (i) the origin is shifted to the point (1, 3).

Solution: We have already derived the formulas for shifting of origin. Using these formulas, we get x = X + h, y = Y + k 

Solving further, 

3 = X + 1 ⇒ X = 2 

4 = Y + 3 ⇒ Y = 1 

Hence, the coordinates concerning the shifted origin are (2, 1).

Example 3: To what point should the origin be shifted so that the equation x2+y2– 4x + 6y – 4 = 0 becomes free of the first degree terms (i.e., 4x and 6y)? 

Solution: Let the new origin be at point O’ (h, k). 

Using the formulas for new coordinates are shifting of origin and replacing x by (X + h) and y by (Y + k), we get: 

x2+y2 + X(-4 + 2h) + Y(6 + 2k) + h2 + k2 – 4h + 6k – 4 = 0 

By first-degree terms, we refer to those terms of x whose power is 1. As per the question, we wish to delete the first-degree terms. Hence, we equate the first-degree terms’ coefficients to zero. 

We have (–4 + 2h)=0 and (6 + 2k)=0. We get h = 2 and k = -3. 

Hence, the new equation will be x2+y2 – 17 = 0.

How can I better prepare for CBSE Class 11 Shift of origin theory?

• Make a timetable before beginning with the topic. The IIT-JEE Mains and Advanced syllabus are very detailed, and the schedule should be made with proper time allotment.
• The shift of origin may seem tricky if only read and understood. Making physical notes can be beneficial as it helps retain the concepts in your mind for longer.
• The shift of origin may require some practice on the graph. Graphical representation makes understanding easier.
• Practice as many quizzes, mock tests, and previous years’ papers as possible. This gives an idea of the weightage and essential topics from CBSE Class 11: Shift of Origin.
• The shift of origin includes word problems, which can be tricky sometimes. Read the problem carefully and take frequent breaks while studying to avoid confusion. 

Conclusion:

The shift of Origin is an important concept. To solve questions related to analytic geometry, it is vital to understand the definition, significance, derivations, and related examples. This blog explains all the basics of Shift of Origin in a nutshell.