General Equation Of A Line

We are given the equation:

x%5E2%20-%20y%5E2%20%2B%202y%20-%201%20%3D%200.

Step 1: Completing the Square
To eliminate the linear

y

-term, complete the square:

x%5E2%20-%20(y%5E2%20-%202y)%20-%201%20%3D%200.

Rewriting

y%5E2%20-%202y

y%5E2%20-%202y%20%3D%20(y%20-%201)%5E2%20-%201.

Thus, substituting back:

x%5E2%20-%20%5Cleft(%20(y%20-%201)%5E2%20-%201%20%5Cright)%20-%201%20%3D%200.

x%5E2%20-%20(y%20-%201)%5E2%20%2B%201%20-%201%20%3D%200.

x%5E2%20-%20(y%20-%201)%5E2%20%3D%200.

Step 2: Shifting the Origin
Introduce a new coordinate system

Y%20%3D%20y%20-%201

, where the origin is shifted to

(0%2C1)

. The transformed equation becomes:

x%5E2%20-%20Y%5E2%20%3D%200.

Since

Y%20%3D%20y%20-%201

, we conclude:

x%5E2%20-%20y%5E2%20%3D%200.

Thus, the transformed equation is:

%5Cboxed%7Bx%5E2%20-%20y%5E2%20%3D%200.%7D

Chapter Questions
1 MCQs