# How do use the method of translation of axes to sketch the curve 9x^2-4y^2-36x-24y-36=0?

Jul 8, 2017

#### Explanation:

$9 {x}^{2} - 4 {y}^{2} - 36 x - 24 y - 36 = 0$ can be written as

$9 \left({x}^{2} - 4 x + 4\right) - 36 - 4 \left({y}^{2} + 6 y + 9\right) + 36 = 36$

or $9 {\left(x - 2\right)}^{2} - 4 {\left(y + 3\right)}^{2} = 36$

or${\left(x - 2\right)}^{2} / {2}^{2} - {\left(y + 3\right)}^{2} / {3}^{2} = 1$

Hence, let us move $\left(0 , 0\right)$ to $\left(2 , - 3\right)$, i.e. $x \to x ' + 2$ and $y \to y ' - 3$, which makes $x = 2$ as new $y$-axis and $y = - 3$ as new $x$-axis and our equation becomes

$\frac{x {'}^{2}}{2} ^ 2 - \frac{y {'}^{2}}{3} ^ 2 = 1$, the equation of a hyperbola.

graph{9x^2-4y^2-36x-24y-36=0 [-8.59, 11.41, -7.96, 2.04]}

graph{x^2/4-y^2/9=1 [-10.59, 9.41, -5.2, 4.8]}