Find the Properties 9x^2-4y^2=36

Problem

9*x2−4*y2=36

Solution

1. Standardize the equation by dividing both sides by 36 to set the right side equal to 1

(9*x2)/36−(4*y2)/36=36/36

(x2)/4−(y2)/9=1

2. Identify the conic section as a horizontal hyperbola because it follows the form (x2)/(a2)−(y2)/(b2)=1

a2=4⇒a=2

b2=9⇒b=3

3. Determine the center by observing that x and y are not shifted.

Center=(0,0)

4. Calculate the focal distance c using the relationship c2=a2+b2

c2=4+9=13

c=√(,13)

5. Identify the vertices by moving a units from the center along the transverse axis (x-axis).

Vertices=(±2,0)

6. Identify the foci by moving c units from the center along the transverse axis.

Foci=(±√(,13),0)

7. Determine the asymptotes using the formula y=±b/a*x

y=±3/2*x

Final Answer

Properties of *9*x2−4*y2=36:Center *(0,0), Vertices *(±2,0), Foci *(±√(,13),0), Asymptotes *y=±3/2*x

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