Write the equation of a hyperbola with vertices at (1,1) and (9, 1) and foci at (0, 1) and (10, 1).

Standard form of equation for a hyperbola with horizontal transverse axis:

(x-h)^2/a^2-(y-k)^2/b^2=1, (h,k)=(x,y) coordinates of center

For given hyperbola:

x-coordinate of center=(9+1)/2=5 (use midpoint formula)

y-coordinate of center=1

center: (5,1)

length of horizontal transverse axis=8 (from 1 to 9)=2a

a=4

a^2=16

..

Foci

2c=9 ( from 1 to 10)

c=4.5

..

c^2=a^2+b^2

b^2=c^2-a^2=20.25-16=4.25

..

Equation of hyperbola:

(x-5)^2/16-(y-1)^2/4.25=1

hello :

by applying the distributive property :

6 × 4.3 = 6 × ( 4 + 0.3 )

            =   (6 ×   4) + (6 ×   0.3)

            = 24 + 1.8

6 × 4.3 = 25.8

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

  region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

  region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true