Write the equation of a hyperbola with foci at (-1, 1) and (5, 1) and vertices at (0, 1) and (4, 1).

Answer 1

The standard form of the hyperbola is 

The vertices of the hyperbola are (0, 1) and (4, 1) and its foci are (-1, 1) and (5, 1).

The standard form of the equation of a hyperbola with center (h, k) (where a and b are not equals to 0) is (x - h)²/a² - (y - k)²/b² = 1 (Transverse axis is horizontal) Since the y - coordinate is constant in the vertices and foci.

The center of the hyperbola lies at the midpoint of its vertic

The vertices and foci are, respectively a and c units from the center (h, k) and the relation between a, band c is b² = c² - a².

 

The center (h, k) = [ (x₁ + x₂)/2, (y₁ + y₂)/2 ] = [ (0+ 4)/2, (1 + 1)/2 ] = (2, 1).

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To find the value of a, 

2a = distance between vertices

2a = sqrt((4-(0))^2+(1-1)^2)

    = 4

a=2

 

To find the value of c, 

2c = distance between foci

2c = sqrt((5-(-1))^2+(1-1)^2)

    = 6

c=3

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To find the value of b, 

b² = c² - a².

b^2 =9-4

b^2 =5

The equation of the hyperbola is

(x - 2)²/4 - (y- 1)²/5 = 1.

 

Answer 2

The vertices of the hyperbola are (0, 1) and (4, 1) and its foci are (-1, 1) and (5, 1).

Since the y - coordinate is constant in the vertices and foci.

This is horizontal hyperbola.

standard form of hyperbola

"a " is the number in the denominator of the positive term

If the x -term is positive, then the hyperbola is horizontal

a = semi-transverse axis , b = semi-conjugate axis

center: (h, k ) Vertices: (h + a, k ), (h - a, k )

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

 Asympototes of hyperbola is

So The y coodinate of the center of hyperbola is  1.

vertices: (0,1) and (4,1)

h + a = 0 ----> (1)

h - a = 4 ------> (2)

Add the equations (1) & (2).

h + a + h - a = 0 + 4

2h = 4

h = 4/2 = 2

So x coordinate of center is 2.

Substitute the h  value in (1),

2 + a = 0

a = - 2 (Since a is positive )

Therfore a  =  2

foci: (-1,1) and (5,1)

h + c = -1

2 + c = -1

c = -3

c² = a² + b²

(-3)² = (2)² + b²

9 - 4 = b²

b = √5

Graph of hyprbola

Center is (h, k ) = (2,1)

a = 2 and b = √5 = 2.236.

asympototes of the hyperbola

a = 2 and b = √5 = 2.236.

Substitute the (h ,k) and (a, b) in standard equation .

the hyperbola is

Graph

Draw the coordinate plane.

Plot the center of hyperbola (2,1).

To graph the hyperbola go 2.236 units up and down from center point and 2 units left and right from center point.

Use these points to draw a rectangle .

Draw diagonal lines through the center and the corner of the rectangle. These are asymptotes.

Plot the vertices and foci of hyperbola.

Draw the curves, beginning at each vertex separately, that hug the asymptotes the farther away from the vertices the curve gets.

The graph approaches the asymptotes but never actually touches them.

The above figure shows the hyperbola.