# How do use the method of translation of axes to sketch the curve x^2-y^2+4x-2y+2=0?

Jan 19, 2017

${X}^{2} - {Y}^{2} = 1$

#### Explanation:

Begin by completing the square

${x}^{2} - {y}^{2} + 4 x - 2 y + 2 = 0$

$\left({x}^{2} + 4 x\right) - \left({y}^{2} + 2 y\right) + 2 = 0$

$\left({\left(x + 2\right)}^{2} - 4\right) - \left({\left(y + 1\right)}^{2} - 1\right) + 2 = 0$

${\left(x + 2\right)}^{2} - 4 - {\left(y + 1\right)}^{2} + 1 + 2 = 0$

$\implies {\left(x + 2\right)}^{2} - {\left(y + 1\right)}^{2} = 1$

IOW a hyperbola based around $\left(- 2 , - 1\right)$

Or in our new co-ordinate system where $X = x + 2 , Y = y + 1$

$\implies {X}^{2} - {Y}^{2} = 1$