Write the general conic form equation using the vertices and perimeter of central rectangle below

Question

Vertices (0,-1), (-20, -1)

Perimeter of Central Rectangle = 72

Answer 1

The conic section here is a hyperbola. From the given vertices we can determine the center is at (-10, -1), hence for C(h, k) we have h = -10 and k = -1.

The length of the transverse axis is the distance between the two vertices = 20 = 2a so a=10.

Since the transverse axis is parallel to the x-axis, the standard form of the equation for a hyperbola is

(x-h)²/a² - (y-k)²/b² = 1

We have a, but need to determine b. Since the perimeter of the central rectangle is given at 72 we can calculate b.

2a + 2b = 72

20 + 2b = 72

2b = 52

b = 26.

Now that we have our a, b, h, and k, we can write the general formula:

(x+10)²/(10)² - (y+1)²/(26)² = 1 or (x+10)²/(100) - (y+1)²/(676) = 1