SOLUTION: Classify the conic section: 1. x2 + y2 = 36 2. x2 - y2 = 36 3. 5x2 + 9y2 = 45 4. x2 + y2 = 121 5. y = 17 6. What is the standard equation

Classify the conic section:

1. x² + y² = 36  <-- circle since x² and y² have the SAME COEFFICIENT, 1,
                     when on the same side of the equation.      

2. x² - y² = 36  <-- hyperbola since x² and y² have coefficients with
                     OPPOSITE SIGNS when on the same side of the equation.

 

3. 5x² + 9y² = 45 <-- ellipse since x² and y² have different coefficients 
                      with THE SAME SIGN when on the same side of the 
                      equation.

 4. x² + y² = 121  <-- circle since x² and y² have the SAME COEFFICIENT, 1,
                       when on the same side of the equation.  

 5. y = 17         <-- line because it has NO x² or y² terms.

 

6. What is the standard equation of the circle with radius 4, and center (0,0)?

Start woth 
   
   (x-h)²+(y-k)²=r² where (h,k) is the center and r is the radius

   (x-0)²+(y-0)²=4²

   x² + y² = 16

7. What is the standard equation of the circle with radius 5, and center (-5,-1)?

Start woth 
   
   (x-h)²+(y-k)²=r² where (h,k) is the center and r is the radius

   (x-(-5))²+(y-(-1))²=5²

     (x+5)² + (y+1)² = 25
 

8. Name the center and radius of the circle whose equation is x² + y² = 49.

   The reverse of the other problems:

   Write it as

   (x-0)²+(y-0)²=7²

   Compare that to

   (x-h)²+(y-k)²=r²

   So the center is (h,k) = (0,0,) and its radius is 7 

 

9. What are the vertices of the ellipse whose equation is (x²)/25 + (y²)/9 = 1?

   The ellipse that has equation 

   

   Since a > b, and a is uder x, the ellipse looks like this 

   has vertices (h+a,k) and (h-a,k).  The center is (h,k)

   Write your equation as

   

   Then the vertices are (h+a,k) and (h-a,k) which are
                         (0+5,0) and (0-5,0)
                           (5,0) and (-5,0)
          

10. What are the vertices of the ellipse whose equation is (x²)/9 + (y²)/36 = 1?

   The ellipse that has equation 

   

   Since a > b, and a is under y, the ellipse looks like this  
   
has vertices (h,k+a) and (h,k-a).  The center is (h,k)

   Write your equation as

   

   Then the vertices are (h,k+a) and (h,k-a) which are
                         (0,0+6) and (0,0-6)
                          (0,-6) and (0,6)


11. What is the center of the ellipse whose equation is (x²)/25 + (y²)/4 = 1?

    Write it as

     

     compare to

     

     So center = (h,k) = (0,0) 


12. What is the center of the ellipse whose equation is (x+1)²/4 + (y²)/1 = 1?

 
   The ellipse that has equation 

   

   Since a > b, and a is under x, the ellipse looks like this 

   It has center is (h,k)

   Write your equation as

   

   Compare it to

   

   and see x+1 = x-h   and   y-k = y-0
             1 = -h           -k = 0
            -1 = h             k = 0

   So the center = (h,k) = (-1,0)

13. What is the center of the hyperbola whose equation is (x²) - (y²) = 1?

   The hyperbola that has equation 

   

   Since a is uder x, the hyperbola looks like this )(

   It has has center is (h,k)

   Write your equation as

   

   Compare to

   

   and h=0, k=0, so its center is (h,k) = (0,0) 



14. What is the center of the hyperbola whose equation is (x²)/4 - (y²)/9 = 1?

   The hyperbola that has equation 

   

   Since a is uder x, the hyperbola looks like this )(
   It has has center is (h,k)

   Write your equation as

   

   Compare to

   

   and h=0, k=0, so its center is (h,k) = (0,0)

 

15. What is the center of the hyperbola whose equation is [(x+2)²/9] - [(y-2)²/16] = 1?

   The hyperbola that has equation 

   

   Since a is under x, the hyperbola looks like this )(
   It has has center is (h,k)

   Write your equation as

   

   Compare to

   

   x-h = x+2       y-k = y-2
    -h = 2          -k = -2
     h = -2          k = 2

     So the center is (h,k) = (-2,2)

 16. How does the graph of the hyperbola whose equation is (x²)/9 - (y²)/25 = 1 open?

   The hyperbola that has equation 

   

   Since a is under x, the hyperbola looks like this )(,

   so it opens right and left

 

17. How does the graph of the hyperbola whose equation is (y²)/4 - (x²)/9 = 1
 open?

   The hyperbola that has equation 

   

   Since a is under y, the hyperbola looks like this 
   so it opens up and down.

 18. Classify the conic section of the equation x² + 2x + y² + 6y = 15.

Circle since x² and y² have the SAME COEFFICIENT, 1, when on the same 
side of the equation.

 19. Classify the conic section of the equation 4x² + 9y² - 8x - 18y - 19 = 0.

Ellipse since x² and y² have different coefficients with THE SAME SIGNS when
on the same side of the equation.

20. Classify the conic section of the equation 9x² - 3 = 18x + 4y.

Parabola since there is only one term that has a variable squared.

Edwin