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