Find the Properties 36y^2-64x^2-144y-128x-2224=0

Problem

36*y2−64*x2−144*y−128*x−2224=0

Solution

1. Group the terms by variable and move the constant to the other side of the equation.

36*y2−144*y−64*x2−128*x=2224

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

36*(y2−4*y)−64*(x2+2*x)=2224

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

36*(y2−4*y+4)−64*(x2+2*x+1)=2224+36*(4)−64*(1)

36*(y−2)2−64*(x+1)2=2304

4. Divide both sides by 2304 to put the equation in standard form.

(36*(y−2)2)/2304−(64*(x+1)2)/2304=1

((y−2)2)/64−((x+1)2)/36=1

5. Identify the type of conic section and its parameters. This is a vertical hyperbola with center (h,k)=(−1,2) a2=64 and b2=36

a=8

b=6

6. Calculate the distance to the foci using c2=a2+b2

c2=64+36=100

c=10

7. Determine the properties based on the standard form.

• Center: (−1,2)

• Vertices: (−1,2±8)⇒(−1,10),(−1,−6)

• Foci: (−1,2±10)⇒(−1,12),(−1,−8)

• Asymptotes: y−2=±8/6*(x+1)⇒y−2=±4/3*(x+1)

Final Answer

((y−2)2)/64−((x+1)2)/36=1

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