Question

How do you find the standard form of the equation of the hyperbola given the properties foci $( \pm 5,0)$ , length of the conjugate axis $6$ ?

Answer 1

For solving this particular question , we first define the general form of equation , then we compare the given foci with the general form of foci , we know that the length of the conjugate axis is given by $2b$ , substituting all we will get the desired result.

Complete step by step solution:
We have the foci $( \pm 5,0)$ , therefore we can say that the hyperbola is the horizontal transverse axis type hyperbola ,
The general equation is given by ,
$\dfrac{{{{(x - h)}^2}}}{{{a^2}}} - \dfrac{{{{(y - k)}^2}}}{{{b^2}}} = 1......(1)$
Now, we have the length of the conjugate axis that is given by $2b = 6$,
Therefore, we get $b = 3..............(2)$
Now, we also know that the general forms of foci is given by ,
$\left( {h - \sqrt {{a^2} + {b^2}} ,k} \right)$ and $\left( {h + \sqrt {{a^2} + {b^2}} ,k} \right)$
By equating general form with the given foci that is $( - 5,0)$ and $(5,0)$.
We will get ,
$k = 0....................(3)$ ,
$h - \sqrt {{a^2} + {b^2}} = - 5....................(4)$,
$h + \sqrt {{a^2} + {b^2}} = 5......................(5)$ .
Now adding $(4)$and $(5)$ , we will get ,
$h = 0..................(6)$
Now, substitute $(2)$ and $(6)$ in equation $(5)$ ,
We will get ,
$\Rightarrow h + \sqrt {{a^2} + {b^2}} = 5$
$\Rightarrow 0 + \sqrt {{a^2} + {3^2}} = 5$
Taking square both the side ,
$\Rightarrow {a^2} + {3^2} = {5^2}$
Subtract ${3^2}$ from both the side ,
$\Rightarrow {a^2} = {5^2} - {3^2}$
$\Rightarrow {a^2} = {4^2}$
Now, taking square root both the side,
$\Rightarrow a = 4......................(7)$
Now, substitute $(2),(3),(6)$ and $(7)$ in equation $(1)$ ,
We will get ,
$\Rightarrow \dfrac{{{{(x - h)}^2}}}{{{a^2}}} - \dfrac{{{{(y - k)}^2}}}{{{b^2}}} = 1 \\\ \Rightarrow \dfrac{{{{(x - 0)}^2}}}{{{4^2}}} - \dfrac{{{{(y - 0)}^2}}}{{{3^2}}} = 1 \\\$
Hence we get the required equation .

Note: The important thing to recollect about any equation is that the ‘equals’ sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equal to what’s on the right-hand side.