Answer to Question #171144 in Analytic Geometry for Jon jay mendoza

Question

A. Given the following information about the hyperbola, find both its standard and general form:

1. FOCI: (-4, 0) (4, 0); VERTICES: (-3, 0) (3, 0)

2. FOCI: (-3, -3) (-3, 13); VERTICES: (-3, 0) (-3, 10)

B. Given the standard form of the equation of hyperbola, find the following, then sketch the graph:

A. orientation

B. opening of branches

C. center

D. transverse axis

E. conjugate axis

F. distance of the foci

G. vertices

H. foci

I. co-vertices

J. asymptotes

1.(x+4)^2/9 − (y+3)^2/4 = 1

2.(y+2)^2/36 − (x−1)^2/49 = 1

C. Convert each general form to standard form of the equation of hyperbola: (10 points)

1. 9y^2 − 4x^2 − 18y + 24x − 63 = 0

2. 9x^2 - 4y^2 - 90x + 32y - 163 = 0

Answer 1

A. 1. Basic “horizontal” hyperbola:

Equation: $\dfrac{x^2}{a^2 }-\dfrac{y^2 }{b^2 }=1$

$c^2=a^2+b^2$

Foci: $(-c, 0), (c, 0)$

Vertices: $(-a, 0), (a, 0)$

$a=3, c=4$

$$b^2=4^2 -3^2=7$$

The standard form of an equation of a hyperbola

$$\dfrac{x^2}{9 }-\dfrac{y^2 }{7 }=1$$

The general form of an equation of a hyperbola

$$6x^2-9y^2-63=0$$

2. The center is $(h, k)$ and the vertices are $(h, k\pm b)$

Equation: $\dfrac{(y-k)^2}{b^2 }-\dfrac{(x-h)^2 }{a^2 }=1$

Foci: $(h, k-c), (h, k+c)$

$h=-3, b=5, c=8, k=5$

$$a^2=8^2-5^2=39$$

The standard form of an equation of a hyperbola

$$\dfrac{(y-5)^2}{25}-\dfrac{(x+3)^2}{39 }=1$$

The general form of an equation of a hyperbola

$$-25x^2-150x+39y^2-390y-225=0$$

B.

1.

$$\dfrac{(x+4)^2}{9}-\dfrac{(y+3)^2}{4 }=1$$

A. Horizontal oriented hyperbola 

B. Opens left and right.

C. Center $(-4,-3).$

D. Major (transverse) axis length: 6

E. Minor (conjugate) axis length: 4

F. $c=\sqrt{13}$

Focal Parameter: $\dfrac{4\sqrt{13}}{13}$

G. Vertices: $(-7, -3), (-1, -3)$

H. Foci: $(-4-\sqrt{13}, -3), (-4+\sqrt{13}, -3)$

I. Co-vertices: $(-4, -5), (-4, -1)$

J. Asymptotes:$y=-\dfrac{2}{3}x-\dfrac{17}{3}, y=\dfrac{2}{3}x-\dfrac{1}{3}$

2.

$$\dfrac{(y+2)^2}{36}-\dfrac{(x-1)^2}{49 }=1$$

A. Verticaloriented hyperbola 

B. Opens upward and downward.

C. Center $(1,-2).$

D. Major (transverse) axis length: 12

E. Minor (conjugate) axis length: 14

F. $c=\sqrt{85}$

Focal Parameter: $\dfrac{49\sqrt{85}}{85}$

G. Vertices: $(1, -8), (1, 4)$

H. Foci: $(1,-2-\sqrt{85}), (1,-2+\sqrt{85})$

I. Co-vertices: $(-6, -2), (8, -2)$

J. Asymptotes:$y=-\dfrac{6}{7}x-\dfrac{8}{7}, y=\dfrac{6}{7}x-\dfrac{20}{7}$

C.

1.

$$9y^2-4x^2-18y+24x-63=0$$

$$9(y^2-2y+1)-4(x^2-6x+9)=36$$

$$\dfrac{(y-1)^2}{4}-\dfrac{(x-3)^2}{9}=1$$

2.

$$9x^2-4y^2-90x+32y-163=0$$

$$9(x^2-10x+25)-4(y^2-8y+16)=324$$

$$\dfrac{(x-5)^2}{36}-\dfrac{(y-4)^2}{81}=1$$