Find the Properties 16x^2-9y^2-64x-18y-89=0

Problem

16*x2−9*y2−64*x−18*y−89=0

Solution

1. Group the x and y terms together and move the constant to the other side of the equation.

16*x2−64*x−9*y2−18*y=89

2. Factor out the leading coefficients from the x and y groups.

16*(x2−4*x)−9*(y2+2*y)=89

3. Complete the square for both the x and y expressions by adding (b/2)2 inside the parentheses and balancing the equation on the right side.

16*(x2−4*x+4)−9*(y2+2*y+1)=89+16*(4)−9*(1)

4. Simplify the right side and write the trinomials as squared binomials.

16*(x−2)2−9*(y+1)2=144

5. Divide both sides by 144 to put the equation into the standard form of a hyperbola.

((x−2)2)/9−((y+1)2)/16=1

6. Identify the center (h,k) and the values of a2 and b2

(h,k)=(2,−1)

a2=9⇒a=3

b2=16⇒b=4

7. Calculate the distance to the foci c using the relation c2=a2+b2

c2=9+16=25⇒c=5

8. Determine the properties based on the horizontal transverse axis.

Vertices: *(h±a,k)⇒(2±3,−1)⇒(−1,−1),(5,−1)

Foci: *(h±c,k)⇒(2±5,−1)⇒(−3,−1),(7,−1)

Asymptotes: *y−k=±b/a*(x−h)⇒y+1=±4/3*(x−2)

Final Answer

Center: *(2,−1), Vertices: *(−1,−1),(5,−1), Foci: *(−3,−1),(7,−1), Asymptotes: *y+1=±4/3*(x−2)

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