Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $9 y^{2} - 25 x^{2} = 225$

검증된 단계별 안내

Rewrite the given equation in standard form by dividing both sides by 225: $\frac{9y^2}{225} - \frac{25x^2}{225} = \frac{225}{225}$, which simplifies to $\frac{y^2}{25} - \frac{x^2}{9} = 1$.

Identify the values of $a^2$ and $b^2$ from the standard form: here, $a^2 = 25$ and $b^2 = 9$. Since the $y^2$ term is positive, the hyperbola opens vertically.

Find the vertices using $a$: vertices are located at $(0, \pm a)$, so $(0, \pm 5)$.

Calculate the foci using $c^2 = a^2 + b^2$: find $c$ by $c = \sqrt{25 + 9}$, then the foci are at $(0, \pm c)$.

Write the equations of the asymptotes using the formula for vertical hyperbolas: $y = \pm \frac{a}{b} x$, so the asymptotes are $y = \pm \frac{5}{3} x$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Form of a Hyperbola

A hyperbola's equation can be written in standard form as either (y-k)^2/a^2 - (x-h)^2/b^2 = 1 or (x-h)^2/a^2 - (y-k)^2/b^2 = 1. This form helps identify the center, vertices, and orientation of the hyperbola. In the given equation, rewriting it to standard form reveals these key features.

Vertices and Foci of a Hyperbola

Vertices are points where the hyperbola intersects its principal axis, located a distance 'a' from the center. Foci lie further out at a distance 'c', where c^2 = a^2 + b^2. Knowing how to find vertices and foci is essential for graphing and understanding the shape of the hyperbola.

Equations of Asymptotes

Asymptotes are lines that the hyperbola approaches but never touches, guiding its shape. For hyperbolas centered at (h,k), the asymptotes have equations y - k = ±(a/b)(x - h) or y - k = ±(b/a)(x - h), depending on orientation. Finding these lines helps in sketching an accurate graph.

Use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9y2−25x2=2259y^2−25x^2=225