Question

Hello! Could anyone help me with this question? I don't understand it.

Question: The equation x^2/24^2-y^/--^2 represents a hyperbola centered at the origin with a directix of x=576/26.

The positive value _____ correctly fills in the blank in the equation.

I'll give 50 points for this question ^^

Answer 1

10

Explanation:

EDGE ............................................................................................

Answer 2

`\bf \textit{hyperbolas, horizontal traverse axis } \\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad √( a ^2 + b ^2)\\ \textit{asymptotes}\quad y= k\pm \cfrac{b}{a}(x- h)\\[0.8em] directrix\qquad x=\cfrac{a^2}{c} \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}`

`\bf \cfrac{x^2}{24^2}-\cfrac{y^2}{b^2}=1\implies \cfrac{(x-0)^2}{24^2}-\cfrac{(y-0)^2}{b^2}=1 \\\\\\ \stackrel{\textit{we know that}}{x=\cfrac{a^2}{c}}\implies \cfrac{576}{26}=\cfrac{24^2}{c}\implies \cfrac{576}{26}=\cfrac{576}{c}\implies c=\cfrac{576\cdot 26}{576}\implies \boxed{c=26} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{\textit{we also know that}}{c=√(a^2+b^2)}\implies c^2-a^2=b^2\implies √(c^2-a^2)=b~~ \begin{cases} c=26\\ a=24 \end{cases}`

`\bf √(26^2-24^2)=b\implies √(100)=b\implies \boxed{10=b} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \cfrac{x^2}{24^2}-\cfrac{y^2}{10^2}=1~\hfill`