# Is the following equation an ellipse, circle, parabola, or hyperbola 4x^2 + 9y^2 - 16x +18y -11 = 0?

May 11, 2018

It is an ellipse with center at $\left(2 , 1\right)$, major axis parallel to $x$-axis is $6$ and minor axis parallel to $y$-axis is $4$.

#### Explanation:

Such equation of ellipse is of the type ${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$, for hyperbola it is of type ${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$ and for parabola it is $\left(y - k\right) = a {\left(x - h\right)}^{2}$ or $\left(x - h\right) = a {\left(y - k\right)}^{2}$.

As we have coefficients of both ${x}^{2}$ and ${y}^{2}$ of same sign, it appears to be an ellipse. Let us convert this into desired form.

$4 {x}^{2} + 9 {y}^{2} - 16 x + 18 y - 11 = 0$ can be written as

$4 {x}^{2} - 16 x + 9 {y}^{2} + 18 y - 11 = 0$

or $4 \left({x}^{2} - 4 x + 4\right) + 9 \left({y}^{2} + 2 y + 1\right) = 11 + 16 + 9$

or $4 {\left(x - 2\right)}^{2} + 9 {\left(y + 1\right)}^{2} = 36$

or ${\left(x - 2\right)}^{2} / 9 + {\left(y + 1\right)}^{2} / 4 = 1$

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

which is an ellipse with center at $\left(2 , 1\right)$, major axis parallel to $x$-axis is $6$ and minor axis parallel to $y$-axis is $4$ (double of $a$ and $b$ respectively).

graph{(4x^2+9y^2-16x+18y-11)((x-2)^2+(y+1)^2-0.01)=0 [-3.29, 6.71, -3.36, 1.64]}