Question

What does the equation (a − b) (x² + y²) −2abx = 0 become if the origin is shifted to the point $\left(\frac{ab}{a-b},0\right)$ without rotation?

Answer 1

Substituting $x=X+\frac{ab}{a-b},y=Y+0$ in the given equation, we get:

$$\begin{gathered} \left(a-b\right)\left[{\left(X+\frac{ab}{a-b}\right)}^2+Y^2\right]-2ab\times \left(X+\frac{ab}{a-b}\right)=0 \\ \Rightarrow \left(a-b\right)\left(X^2+\frac{a^2{b}^2}{{\left(a-b\right)}^2}+\frac{2abX}{a-b}+Y^2\right)-2abX-\frac{2a^2{b}^2}{a-b}=0 \\ \Rightarrow \left(a-b\right)\left(X^2+Y^2\right)+\frac{a^2{b}^2}{a-b}+2abX-2abX-\frac{2a^2{b}^2}{a-b}=0 \\ \Rightarrow \left(a-b\right)\left(X^2+Y^2\right)-\frac{a^2{b}^2}{a-b}=0 \\ \Rightarrow {\left(a-b\right)}^2\left(X^2+Y^2\right)=a^2{b}^2 \end{gathered}$$

Hence, the transformed equation is ${\left(a-b\right)}^2\left(X^2+Y^2\right)=a^2{b}^2$.