Question

How do you write the equation of the hyperbola given foci: $\left( {0, - 5} \right)$, $\left( {0,5} \right)$ and vertices $\left( {0, - 3} \right)$, $\left( {0,3} \right)$.

Answer 1

Given the coordinates of the foci and the vertices, we have to find the equation of the hyperbola. First, we have to determine that the transverse axis lies on $x$ or $y$-axis. Then, we will find the value of the centre h and k. We will get the value of variable b by applying the Pythagorean Theorem to the vertices and foci of the hyperbola. Then, we will find the equation of the hyperbola by substituting all the values into the general equation of the hyperbola.

Formula used: The general equation of the hyperbola where the transverse axis is on the $y$-axis is given as:
$\dfrac{{{y^2}}}{{{a^2}}} - \dfrac{{{x^2}}}{{{b^2}}} = 1$
Where the vertices are $\left( {0, \pm a} \right)$ and the foci are $\left( {0, \pm c} \right)$
The Pythagorean Theorem states that the sum of squares of two sides is equal to the square of the third side.

Complete step-by-step solution:
First, determine the $x$-coordinate of the foci and vertices.
$\Rightarrow x = 0$
Here, the $x$-coordinate is zero which means the vertices of the parabola are on the $y$-axis and the coordinates of the centre is $\left( {0,0} \right)$ which means the value of $h = 0$ and $k = 0$
Determine the value of $a$ by comparing the coordinates of the vertices with $\left( {0, \pm a} \right)$
$\Rightarrow a = 3$
Determine the value of $c$ by comparing the coordinates of the vertices with $\left( {0, \pm c} \right)$
$\Rightarrow c = 5$
Now, we will determine the value of ${b^2}$ by substituting the values of $a$and $c$into the Pythagorean theorem.
$\Rightarrow {a^2} + {b^2} = {c^2}$
$\Rightarrow {3^2} + {b^2} = {5^2}$
On simplifying the equation, we get:
$\Rightarrow {b^2} = 25 - 9$
$\Rightarrow {b^2} = 16$
Now, we will find the equation of the hyperbola by substituting the values into the standard equation of the hyperbola.
$\Rightarrow \dfrac{{{y^2}}}{{{3^2}}} - \dfrac{{{x^2}}}{{16}} = 1$
$\Rightarrow \dfrac{{{y^2}}}{9} - \dfrac{{{x^2}}}{{16}} = 1$

Hence the equation of the hyperbola is $\dfrac{{{y^2}}}{9} - \dfrac{{{x^2}}}{{16}} = 1$

Note: In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. So when the foci and the vertices are given, then the first thing is to determine the transverse axis, whether it lies on $x$ or $y$-axis. Then, according to the transverse axis, the general equation of the hyperbola is used.