GRAPH THE HYPERBOLA AND IDENTIFY THE CENTER VERTICES FOCI AND ASYMPTOTES

Graph the hyperbola and identify the center, vertices, slopes of asymptotes and foci.

Problem 1 :

Solution:

a² = 9

a = 3

b² = 16

b = 4

The above hyperbola is symmetric about y-axis.

Center:

(0, 0)

Vertices:

The coordinates of the vertices are (0, ±a)

vertices = (0, ±3)

Foci:

The coordinates of the foci are (0, ±c)

c² = a² + b²

= 9 + 16

c² = 25

c = 5

Foci = (0, ±5)

Asymptotes:

Problem 2 :

4x² - y² = 16

Solution:

4x² - y² = 16

a² = 4

a = 2

b² = 16

b = 4

The above hyperbola is symmetric about x-axis.

Center:

(0, 0)

Vertices:

The coordinates of the vertices are (±a, 0)

vertices = (±2, 0)

Foci:

The coordinates of the vertices are (0, ±c)

c² = a² + b²

= 4 + 16

c² = 20

c = 2√5

Foci = (±2√5, 0)

Asymptotes:

Problem 3 :

4(x - 1)² - 9(y + 2)² = 36

Solution:

4(x - 1)² - 9(y + 2)² = 36

a² = 9

a = 3

b² = 4

b = 2

The above hyperbola is symmetric about x-axis.

Center:

(0, 0)

X = 0 and Y = 0

Substitute X = x - 1 and Y = y + 2

x - 1 = 0 and y + 2 = 0

x = 1 and y = -2

The center is (1, -2).

Vertices:

The coordinates of the vertices are (±a, 0)

A(a, 0) and A'(-a, 0)

A(3, 0) and A'(-3, 0)

The vertices are (4, -2) and (-2, -2).

Foci:

c² = a² + b²

= 9 + 4

c² = 13

c = √13

Here (h, k) = center of the hyperbola

h = 1

k = -2

Foci = (h ± c, k)

= (1 ± √13, -2)

Asymptotes:

Problem 4 :

Solution:

a² = 25

a = 5

b² = 9

b = 3

The above hyperbola is symmetric about y-axis.

Center:

(0, 0)

x - 2 = 0 and y + 1 = 0

x = 2 and y = -1

The center is (2, -1).

Vertices:

The coordinates of the vertices are (0, ±a)

A(0, a) and A'(0, -a)

A(0, 5) and A'(0, -5)

The vertices are (2, 4) and (2, -6).

Foci:

c² = a² + b²

= 25 + 9

c² = 34

c = √4

Here (h, k) = center of the hyperbola

h = 2

k = -1

Foci = (h, k ± c)

= (2, -1 ± √34)

Asymptotes:

Problem 5 :

Solution:

a² = 25

a = 5

b² = 16

b = 4

The above hyperbola is symmetric about y-axis.

Center:

(0, 0)

x - 3 = 0 and y + 2 = 0

x = 3 and y = -2

The center is (3, -2).

Vertices:

The coordinates of the vertices are (0, ±a)

A(0, a) and A'(0, -a)

A(0, 5) and A'(0, -5)

The vertices are (3, 3) and (3, -7).

Foci:

c² = a² + b²

= 25 + 16

c² = 41

c = √41

Here (h, k) = center of the hyperbola

h = 3

k = -2

Foci = (h, k ± c)

= (3, -2 ± √41)

Asymptotes:

Problem 6 :

Solution:

a² = 25

a = 5

b² = 16

b = 4

The above hyperbola is symmetric about x-axis.

Center:

(0, 0)

x - 2 = 0 and y + 3 = 0

x = 2 and y = -3

The center is (2, -3).

Vertices:

The coordinates of the vertices are (±a, 0)

A(a, 0) and A'(-a, 0)

A(5, 0) and A'(-5, 0)

The vertices are (7, -3) and (-3, -3).

Foci:

c² = a² + b²

= 25 + 16

c² = 41

c = √41

Here (h, k) = center of the hyperbola

h = 2

k = -3

Foci = (h, ± c, k)

= (2, ± √41, -3)

Asymptotes: