Determine the coordinates of the vertices of the hyperbola $H$, given by its general equation: $$4x^2 - 3y^2 + 16x + 18y - 23 = 0.$$
Eric solved the example in the following steps:
(1) He converted the general equation of the hyperbola to its standard form: $$\begin{alignat}2 4x^2 - 3y^2 + 16x + 18y - 23 &= 0 \cr 4x^2 + 16x - 3y^2 + 18y - 23 &= 0 \cr 4(x^2 + 4x) - 3(y^2 - 6y) - 23 &= 0 \cr 4(x^2 + 4x + 4 - 4) - 3(y^2 - 6y + 9 - 9) - 23 &= 0 \cr 4\left[(x + 2)^2 - 4\right] - 3\left[(y - 3)^2 - 9\right] - 23 &= 0 \cr 4(x + 2)^2 - 16 - 3(y - 3)^2 + 27 - 23 &= 0 \cr 4(x + 2)^2 - 3(y - 3)^2 - 12 &= 0\quad &&\big/ + 12 \cr 4(x + 2)^2 - 3(y - 3)^2 &= 12\quad &&\big/ : 12 \cr \frac{(x + 2)^2}{3} - \frac{(y - 3)^2}{4} &= 1 \end{alignat}$$
(2) He determined the coordinates of the center, denoted $S$, of the hyperbola: $S=[-2; 3]$.
(3) He determined the lengths of the hyperbola semi-axes:
- $4 > 3 \Rightarrow$ The length of the semi-major axis, denoted $a$, is $2$ units $(a^2 = 4)$.
- The length of the semi-minor axis, denoted $b$, is $\sqrt3$ units $(b^2 = 3)$.
(4) He determined the coordinates of the vertices of the hyperbola. Eric claimed that the major axis of the hyperbola is parallel to the y-axis. Therefore, the vertices of the hyperbola have coordinates $[-2; 3 – 2]$ and $[-2; 3 + 2]$, i.e., $[-2; 1]$ and $[-2; 5]$.
Eric's solution is not correct. Where did Eric make his first mistake?
The first mistake is in step (1). Eric made a mistake in converting the given equation of the hyperbola to its standard form.
The first mistake is in step (2). Eric incorrectly determined the coordinates of the center of the hyperbola.
The first mistake is in step (3). Eric incorrectly determined the lengths of the major and minor semi-axes.
The first mistake is in step (4). Eric incorrectly determined the coordinates of the vertices of the hyperbola.
(1) Given the hyperbola equation $4x^2 – 3y^2 + 16x + 18y – 23 = 0$, we transform it to standard form $H$: $$\frac{(x + 2)^2}{3} –\frac{(y - 3)^2}{4} = 1.$$
(2) The center, $S$, of the hyperbola is: $S=[m; n]=[-2; 3]$.
(3) We find the lengths of the hyperbola major and minor semi-axis, denoted as $a$ and $b$:
Given that $a^2$ is in the hyperbola equation in the fraction not preceded by the negative sign $($-$)$, we have $a^2=3$. The length of the semi-major axis, $a$, is therefore $\sqrt3$ units.
Given that $b^2$ is in a hyperbola equation in the fraction preceded by the negative sign $($-$)$, we have $b^2=4$. The length of the semi-minor axis, $b$, is therefore $2$ units.
(4) The major axis, $a$, of the given hyperbola is parallel to the x-axis. Therefore, the vertices of the hyperbola are: $[m-a;n]=\left[-2-\sqrt3;3\right]$ and $[m+a;n]=\left[-2+\sqrt3;3\right]$.