Find the equation and length of major and minor axes of $\;$ $5x^2 + 9y^2 + 10 x - 36 y - 4 = 0$
Equation of the given ellipse is: $\;$ $5x^2 + 9y^2 + 10 x - 36 y - 4 = 0$
i.e. $\;$ $\left(5x^2 + 10x\right) + \left(9y^2 - 36y\right) = 4$
i.e. $\;$ $5 \left(x^2 + 2x\right) + 9 \left(y^2 - 4y\right) = 4$
i.e. $\;$ $5 \left[\left(x^2 + 2x + 1\right) - 1\right] + 9 \left[\left(y^2 - 4y + 4\right) -4\right] = 4$
i.e. $\;$ $5 \left[\left(x + 1\right)^2 - 1\right] + 9 \left[\left(y - 2\right)^2 - 4\right] = 4$
i.e. $\;$ $5 \left(x + 1\right)^2 + 9 \left(y - 2\right)^2 = 4 + 5 + 36$
i.e. $\;$ $5 \left(x + 1\right)^2 + 9 \left(y - 2\right)^2 = 45$
i.e. $\;$ $\dfrac{\left(x + 1\right)^2}{45 / 5} + \dfrac{\left(y - 2\right)^2}{45 / 9} = 1$
i.e. $\;$ $\dfrac{\left(x + 1\right)^2}{9} + \dfrac{\left(y -2 \right)^2}{5} = 1$ $\;\;\; \cdots \; (1)$
Let $\;$ $X = x + 1$ $\;\;\; \cdots \; (2a)$ $\;$ and $\;$ $Y = y - 2$ $\;\;\; \cdots \; (2b)$
Then, in view of equations $(2a)$ and $(2b)$, equation $(1)$ becomes
$\dfrac{X^2}{9} + \dfrac{Y^2}{5} = 1$ $\;\;\; \cdots \; (3)$
The major axis is along the X axis and the minor axis is along the Y axis.
$\therefore$ $\;$ Equation of major axis is: $\;$ $Y = 0$ $\implies$ $y - 2 = 0$
Equation of minor axis is: $\;$ $X = 0$ $\implies$ $x + 1 = 0$
Comparing equation $(3)$ with the standard equation of ellipse $\;$ $\dfrac{X^2}{a^2} + \dfrac{Y^2}{b^2} = 1$ $\;$ $\left(\text{for } a > b\right)$ $\;$ gives
$a^2 = 9 \implies a = 3$; $\;$ and $\;$ $b^2 = 5 \implies b = \sqrt{5}$
$\therefore$ $\;$ Length of major axis $= 2a = 2 \times 3 = 6$
Length of minor axis $= 2b = 2 \sqrt{5}$