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:
\[\left( a - b \right)\left[ \left( X + \frac{ab}{a - b} \right)^2 + Y^2 \right] - 2ab \times \left( X + \frac{ab}{a - b} \right) = 0\]
\[ \Rightarrow \left( a - b \right)\left( X^2 + \frac{a^2 b^2}{\left( a - b \right)^2} + \frac{2abX}{a - b} + Y^2 \right) - 2abX - \frac{2 a^2 b^2}{a - b} = 0\]
\[ \Rightarrow \left( a - b \right)\left( X^2 + Y^2 \right) + \frac{a^2 b^2}{a - b} + 2abX - 2abX - \frac{2 a^2 b^2}{a - b} = 0\]
\[ \Rightarrow \left( a - b \right)\left( X^2 + Y^2 \right) - \frac{a^2 b^2}{a - b} = 0\]
\[ \Rightarrow \left( a - b \right)^2 \left( X^2 + Y^2 \right) = a^2 b^2\]
Hence, the transformed equation is \[\left( a - b \right)^2 \left( X^2 + Y^2 \right) = a^2 b^2\].