Question

1. determine the center, vertices, and foci for the following hyperbola $\frac{y^{2}}{16}-\frac{x^{2}}{9}=1$. center vertices foci 2. determine the center, vertices, and foci for the following hyperbola $\frac{(x + 1)^{2}}{16}-\frac{(y - 2)^{2}}{4}=1$. center vertices foci 3. determine the center, vertices, and foci for the following hyperbola $\frac{y^{2}}{20}-\frac{(x - 1)^{2}}{16}=1$. center vertices foci

Explanation:

Step1: Recall hyperbola standard - form

The standard form of a vertical hyperbola is $\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1$ with center $(h,k)$, vertices $(h,k\pm a)$ and $c=\sqrt{a^{2}+b^{2}}$, foci $(h,k\pm c)$. The standard form of a horizontal hyperbola is $\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1$ with center $(h,k)$, vertices $(h\pm a,k)$ and $c=\sqrt{a^{2}+b^{2}}$, foci $(h\pm c,k)$.

Step2: Solve for hyperbola $\frac{y^{2}}{16}-\frac{x^{2}}{9}=1$

1. Center: For $\frac{y^{2}}{16}-\frac{x^{2}}{9}=1$, comparing with $\frac{(y - 0)^2}{4^2}-\frac{(x - 0)^2}{3^2}=1$, the center is $(0,0)$.
2. Vertices: Since $a = 4$, the vertices are $(0,0\pm4)=(0,\pm4)$.
3. Foci: Calculate $c=\sqrt{4^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5$. The foci are $(0,0\pm5)=(0,\pm5)$.

Step3: Solve for hyperbola $\frac{(x + 1)^{2}}{16}-\frac{(y - 2)^{2}}{4}=1$

1. Center: Comparing with $\frac{(x -(- 1))^{2}}{4^{2}}-\frac{(y - 2)^{2}}{2^{2}}=1$, the center is $(-1,2)$.
2. Vertices: Since $a = 4$, the vertices are $(-1\pm4,2)$, i.e., $(3,2)$ and $(-5,2)$.
3. Foci: Calculate $c=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}$. The foci are $(-1\pm2\sqrt{5},2)$.

Step4: Solve for hyperbola $\frac{y^{2}}{20}-\frac{(x - 1)^{2}}{16}=1$

1. Center: Comparing with $\frac{(y - 0)^2}{(\sqrt{20})^{2}}-\frac{(x - 1)^{2}}{4^{2}}=1$, the center is $(1,0)$.
2. Vertices: Since $a=\sqrt{20}=2\sqrt{5}$, the vertices are $(1,0\pm2\sqrt{5})$.
3. Foci: Calculate $c=\sqrt{20 + 16}=\sqrt{36}=6$. The foci are $(1,0\pm6)$.

Answer:

1.

• Center: $(0,0)$
• Vertices: $(0,4),(0, - 4)$
• Foci: $(0,5),(0, - 5)$

2.

• Center: $(-1,2)$
• Vertices: $(3,2),(-5,2)$
• Foci: $(-1 + 2\sqrt{5},2),(-1-2\sqrt{5},2)$

3.

• Center: $(1,0)$
• Vertices: $(1,2\sqrt{5}),(1,-2\sqrt{5})$
• Foci: $(1,6),(1,-6)$