#20: CONICS



REVIEW OF BASIC CONIC FACTS



The equation Ax 2 + By 2 + Cx + Dy + E = 0 will graph into a conic or a degenerate conic. The particular conic you get depends upon the coefficients A and B:



CONDITION ON A AND B TYPE OF CONIC
AB = 0 PARABOLA
AB > 0 ELLIPSE
AB <0 HYPERBOLA
The coefficients C and D translate the conic to a new center.


1. Determine what type of conic the following equations represent.

  1. 3x 2 - 4y 2 + 3x - 2y - 4 = 0
  2. 9x + 2y 2 + 3y - 7 = 0
  3. 2x - 3y + x 2 + 4y 2 = 0


PARABOLA ( AB = 0 )



2. HOMEWORK #1: Describe in your own words what you look for in an equation to tell if it will graph into a parabola.



3. To graph the parabola 9x + 2y 2 + 3y - 7 = 0

  1. Solve for the non-squared variable. In this case solve for x.
  2. Now you have the equation in the form x = ay 2 + by +c . Complete the squares and write the equation in the form x - h = a ( y - k ) 2
  3. The vertex of the parabola is (h,k) and the line of symmetry is y = k. Plot these on your graph. Choose a value of y not equal to k, evaluate x at your value of y, and plot the point. Plot the symmetric point on the other side of the line y = k. Fill in the rest of the graph of the parabola through these three points and symmetric to the line y = k.
  4. Note: An alternate method to completing the squares is to make the substitution y = u - b /( 2a ). ( It is easy to remember because it is that part of the quadratic formula without the + squat root ) Simplify you expression and the substitute back in terms of y.


4. HOMEWORK #2: Find the vertex, line of symmetry, and graph of the parabola 4x 2 - 4x + 2y + 5 = 0



ELLIPSE ( AB > 0 )



5. HOMEWORK #3: Describe in your own words what you look for in an equation to tell if it will graph into an ellipse.



6. To graph the ellipse 2x - 3y + x 2 + 4y 2 = 0

  1. Collect the two y terms together, the two x terms together, and bring the constant to the other side of the equation.
  2. Now you have an equation of the form ( a y 2 + b y ) + ( c x 2 + d x ) = e. Complete the squares on the two terms on the left and write the equation in the form a ( y - k ) 2 + c ( x - h ) 2 = f. (Again an alternate method is to make the substitution y = u - b /( 2a ) and x = v - d /( 2c ), simplify and then substitute back in terms of x and y. )
  3. Divide both sides by f to get a 1 on the right hand side. ( If f = 0, then the ellipse is the single point ( h,k ) . If f < 0, then the graph is empty. These are the degenerate cases of the ellipse. )
  4. Now divide the numerator and denominator of the ( y - k ) 2 term by a and the numerator and denominator of the ( x - h ) 2 term by c. Write you equation in the standard form ( y - k ) 2/ m 2 + ( x - h ) 2/ n 2 = 1
  5. To graph the ellipse draw a rectangle centered at ( h,k ) with a vertical dimension of 2m and a horizontal dimension of 2n. Draw your ellipse inside this rectangle tangent at the midpoints on the sides of the rectangle.


7. HOMEWORK #4: Graph the ellipse 2x 2 + 4x + y 2 - 2 = 0.



HYPERBOLA ( AB < 0 )



8. HOMEWORK #5: Describe in your own words what you look for in an equation to tell if it will graph into an hyperbola.



9. To graph the hyperbola 3x 2 - 4y 2 + 3x - 2y - 4 = 0

  1. Collect the two y terms together, the two x terms together, and bring the constant to the other side of the equation.
  2. Now you have an equation of the form ( a y 2 + b y ) + ( c x 2 + d x ) = e. Complete the squares on the two terms on the left and write the equation in the form a ( y - k ) 2 + c ( x - h ) 2 = f. Either a or c will be negative but not both.. (Again an alternate method is to make the substitution y = u - b /( 2a ) and x = v - d /( 2c ), simplify and then substitute back in terms of x and y. )
  3. Divide both sides by f to get a 1 on the right hand side. ( If f = 0, then the graph is the two straight lines y - k = + | a / c| 1 / 2 ( x - h ). This is the degenerate cases of the hyperbola. )
  4. Now divide the numerator and denominator of the ( y - k ) 2 term by | a | and the numerator and denominator of the ( x - h ) 2 term by | c |. Write you equation in the standard form - ( y - k ) 2/ m 2 + ( x - h ) 2/ n 2 = 1 or ( y - k ) 2/ m 2 - ( x - h ) 2/ n 2 = 1.
  5. To graph the hyperbola draw a rectangle centered at ( h,k ) with a vertical dimension of 2m and a horizontal dimension of 2n just like in the ellipse. Now draw the two diagonals in the rectangle and extend them into straight lines. These straight lines are the asymptotes of the hyperbola. If the plus sigh is with the y term then the two halves of the hyperbola are tangent to the midpoint at the top and bottom sides of the rectangle and continue away from the rectangle asymptotic to the diagonal lines. If the plus sigh in with the x term then the two halves of the hyperbola are tangent to the midpoint at the right and left sides of the rectangle and continue away from the rectangle asymptotic to the diagonal lines. Just remember, the hyperbola opens in the direction of the variable with the plus sign.


10. HOMEWORK #6: Graph the hyperbola 2x 2 + 4x - y 2 + 6 = 0.



ROTATING THE AXIS ( AN xy IS PRESENT IN THE CONIC )



The equation of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 still graphs into a conic, but it will be a rotated conic. The graph them by hand it is necessary to get rid of the xy term.

It is possible to tell what type of conic the equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 is without graphing it.



THEOREM: Let Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0. Then

CONDITION TYPE OF CONIC
B 2 - AC = 0 PARABOLA
B 2 - AC > 0 HYPERBOLA
B 2 - AC < 0 ELLIPSE


ALTERNATIVE APPROACH



11. With a graphing utility, you can graph any of the conics by solving for y in terms of x using the quadratic formula if necessary. Graph the equation 2x 2 - 3xy - 2y 2 + 2x - 3y + 5 = 0

  1. Group the equation into three parts; the y 2 term, the y term, and the terms that do not contain y.
  2. Find A, B, and C where A is the coefficient of y 2, B is the coefficient of y which will be an expression involving x, and let C is all the terms that do not contain y.
  3. Insert these values for A, B, and C into the quadratic formula to get two solutions for y
  4. Graph these two solutions

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HOMEWORK ASSIGNMENT:

REQUIRED PROBLEMS #1, 2, 3, 4, 5, 6
SUGGESTED PROBLEMS Section 9.1: 9, 13, 21, 27, 29, 35, 39, 41, 45, 53, 59, 69, 75, 77, 79, 87, 95, 101, 105, 106-112, 114-123


A useful web site:

READING ASSIGNMENT BEFORE NEXT WORKSHEET: Section 9.2

rbyrne@math.cudenver.edu
 ROXANNE BYRNE :UNIVERSITY OF COLORADO AT DENVER: ©:1999, Roxanne Byrne