Question

What is the equation of the following graph?

Answer 1

`((x-1)^2)/(5^2) -((y-2)^2)/(2^2) =1`

Explanation:

Here you are require to find the equation of the hyperbola given that the center (h,k), the coordinates of the vertices and those of the co-vertices can be determined from the diagram given

The sharp turning points of the curves give the vertices at (-4,2) and (6,2)

Joining the vertices with a straight line will form the transverse axis with length 2a .

To find the length of the transverse axis 2a will be ; 6--4=10. 2a=10 hence a=10/2 =5

a=5

Find the center of the hyperbola at (h,k) by finding the intersecting point of the diagonals of the rectangle in the diagram

The center identified will be (h,k) = (1,2)

To find the length of the conjugate axis 2b will be ; the length between points (1,4) and (1,0) which are the coordinates of the co-vertices in the hyperbola. 2b= 4-0=4 , b=4/2 = 2

b=2

The standard equation of the hyperbola with center (h,k) is written as ;

`((x-h)^2)/(a^2) -((y-k)^2)/(b^2) =1`

where (h,k) is center of hyperbola, (h±a,k) is coordinate of the vertices and (h,k±b) are coordinates of co-vertices.

Substitute values of a, b, h, and k in equation as

`((x-1)^2)/(5^2) -((y-2)^2)/(2^2) =1`