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Mathematics Dictionary
Dr. K. G. Shih
Conic Sections
Questions

Answers


Q01. Circles

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Q02. Ellipse

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Q03. Hyperbola

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Q04. Parabola

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Q05. Implicit equation F(x,y) = 0 : The discriminant B^2 - 4*A*C

  • Equation : F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0.
  • B^2 - 4*A*C = 0 : It is a parabola.
  • B^2 - 4*A*C < 0 : It is an ellipse.
  • B^2 - 4*A*C > 0 : It is a hyperbola.
  • A = C : It is a circle.
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Q06. Eliminate x*y terms in F(x,y) = 0

  • Equation : F(x,y) = A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0.
  • Rotation :Transform x & y to u & v by
    • (1) x=u*cos(t)-v*sin(t).
    • (2) y=u*sin(t)+v*cos(t).
  • Equation after rotation : A'*u^2+B'*u*v+C'*v^2+D'*u+E'*v+F'=0.
    • A' = A*cos(t)^2+B*cos(t)*sin(t)-C*sin(t)^2.
    • B' = (C-A)*sin(2*t)+B*cos(2*t).
    • C' = A*sin(t)^2-B*cos(t)*sin(t)+C*cos(t)^2.
    • D' = D*cos(t)+E*sin(t).
    • E' = -D*sin(t)+E*cos(t).
    • F' = F.
  • Eliminate u*v term :
    • Let B'=0.
    • (C-A)*sin(2*t)+B*cos(2*t)=0.
    • Hence tan(2*t)=(A-C)/B.
    • Angle t = arctan((A-C)/B)/2.
  • Chnage A'*u^2 + C'*v^2 + D'*u + E'*v + F' = 0 to A'*((u-h)/a)^2 + B'*((v-k)/b)^2 = m.
    • Since no u*v we can use completing square nethod.
    • Hence we find h and k.
    • We can also find a and b.
    • We can also find vertice and focal length f.
  • Demo Example in MD2002 program 34 12.
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Q07. Conic Sections : Standard forms
  • a. Circle : (x-h)^2 + (y-k)^2 = r^2.
    • Center is at (h,k).
    • Radius is r.
  • b. Ellipse : (x-h)^2/a^2 + (y-k)^2/b^2 = 1.
    • Center is at (h,k).
    • Semi axese are a and b.
    • Focal length f = Sqr(a^2-b^2).
    • if a is greater than b :
      • Principal axis is y = k .
      • Vertice are at U(h-f,k) and V(h+f,k).
    • if a is less than b :
      • Principal axis is x = h .
      • Vertice are at U(h,k-f) and V(h,k+f).
  • c. Hyperbola : (x-h)^2/a^2 - (y-k)^2/b^2 = m and m = 1 or m = -1.
    • Center is at (h,k).
    • Semi axese are a and b.
    • Focal length f = Sqr(a^2+b^2).
    • if m is 1 :
      • Principal axis is y = k .
      • Vertice are at U(h-f,k) and V(h+f,k).
    • if m is -1 :
      • Principal axis is x = h .
      • Vertice are at U(h,k-f) and V(h,k+f).
    • Asymptotes : (y - k) = +(b/a)*(x - h) and (y - k) = +(b/a)*(x - h)
  • d. Parabola : y - k = (x - h)^2/(2*D).
    • Vertex is (h,k).
    • Focus to directrix is D.
    • Focus F is at (h,k+D/2).
    • Principal axis is x = h.
    • Equation of directrix is y = k-D/2.
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Q08. Conic Sections : implicit forms
    1. Circle : a*x^2 + a*y^2 +c*x + d*y + e = 0
    • Using completing square convert it to a*(x-h)^2 + a*(y-k)^2 = m.
    • If m is greater than 0, the locus is a circle.
    • If m is equal to 0, the locus is a point.
    • If m is less than 0, the equation does not exist in real number system.
    2. Ellipse : a*x^2 + b*y^2 +c*x + d*y + e = 0
    • Using completing square convert it to a*(x-h)^2 + b*(y-k)^2 = m.
    • If m is greater than 0, the locus is an ellipse.
    • If m is equal to 0, the locus is a point.
    • If m is less than 0, the equation does not exist in real number system.
    3. Hyperbola : a*x^2 - b*y^2 +c*x + d*y + e = 0
    • Using completing square convert it to a*(x-h)^2 - b*(y-k)^2 = m.
    • If m is greater than 0, the locus is a hyperbola with principal axis y = k.
    • If m is equal to 0, the locus is two lines.
    • If m is less than 0, the locus is a hyperbola with principal axis x = h.
    4. Parabola
    • a. Equation : a*x^2 + c*y + d*x + e = 0
      • Convert to standard form y - k = a*(x - h)^2.
      • D = 1/(2*a)
      • The principal axis is x = h.
      • Vertex is at (h,k)
      • It opens to upward if a is greater than 0.
      • It opens to downward if a is less than 0.
    • a. Equation : a*x^2 + c*y + d*x + e = 0
      • Convert to standard form x - h = a*(y - k)^2.
      • D = 1/(2*a)
      • The principal axis is y = k.
      • Vertex is at (h,k)
      • It opens to right if a is greater than 0.
      • It opens to left if a is less than 0.
Example : Find locus of x^2 + y^2 - 2*x + 2*y - 2 = 0
  • (x^2 - 2*x + 1 - 1) + (y^2 + 2*y + 1 - 1) - 2 = 0.
  • (x - 1)^2 + (y + 1)^ = 2^2.
  • The locus is a circle and center at (1,-1) and radius = 2.
Example : Find locus of x^2 + y^2 - 2*x + 2*y + 2 = 0
  • (x^2 - 2*x + 1 - 1) + (y^2 + 2*y + 1 - 1) + 2 = 0.
  • (x - 1)^2 + (y + 1)^ = 0.
  • The locus is a point at (1,-1).
Example : Find locus of x^2 + y^2 - 2*x + 2*y +6 = 0
  • (x^2 - 2*x + 1 - 1) + (y^2 + 2*y + 1 - 1) + 6 = 0.
  • (x - 1)^2 + (y + 1)^ = -4.
  • The locus does not exist in real number system.
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Q09. Conic Sections in polar form

  • Locus definition
    • Focus F is origin.
    • P is point on locus.
    • PF/PQ = e where Q is point on directrix and PQ perpendicualr to directrix.
    • The locus is
      • Ellipse if e is less than 1.
      • Hyperbola if e is greater than 1.
      • Parabola if e is equal to 1.
  • Equation of locus in polar form
    • R = D*e/(1-e*sin(A))
    • R = D*e/(1+e*sin(A))
    • R = D*e/(1-e*cos(A))
    • R = D*e/(1+e*cos(A))
  • Prove the equation in polar form
    • Since PF/PQ = e.
    • PF is R and PQ is D + x where D is distance from directrix.
    • In polar coordinate x = R*cos(A).
    • Hence R/(D + x) = e.
    • Hence R = e*(D + x) = e*(D + R*cos(A)).
    • Hence R = D*e/(1 - e*cos(A)).
  • Relation with rectangular coordinates
    • e = f/a.
    • D*e = (a - f)*(1 + e) for ellipse R = D*e/(1-e*cos(A)) at A = 180.
    • D*e = (f - a)*(1 + e) for hyperbola R = D*e/(1-e*cos(A)) at A = 180
    • D = 1/2*a for y = a*x^2 + b*x + c.
Diagrams
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Q10. Special forms in polar form

  • Circle : R = sin(A).
  • Circle : R = cos(A).
  • Parabola : R = csc(A/2)^2.
  • Parabola : R = sec(A/2)^2.
Note : The proof can be found in webpage of circle and parabola. Go to Begin

Q11. Special forms in parametric form

  • Circle : x = h + R*cos(t) and y = k + R*sin(t).
  • ellipse : x = h + a*cos(t) and y = k + b*sin(t).
  • hyperbola : x = h + a*sec(t) and y = k + b*tan(t).
Note : The proof can be found in webpage of circle, ellipse, hyperbola.
Diagrams : Patterns of parametric equations
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Q12. Quiz of conic sections

  • 1. Why circle, parabola, ellipse and hyperbola are named as conic sections.
  • 2. Find locus of x^2 + y^2 + 4*x - 4*y + 8 = 0.
  • 3. Find locus of x^2 + y^2 + 4*x - 4*y - 8 = 0.
  • 4. Find locus of x^2 + y^2 + 4*x - 4*y + 10 = 0.
  • 5. The function is y = x^2 - 6*x + 8.
    • a. Find the vertex.
    • b. Find equation of the principal axis.
    • c. Find the distance from focus to the directrix.
    • d. Find the coordinates of focus.
    • e. Find equation of the directrix.
    • f. Find the openning direction of this parabola.
  • 6. Give the standard form of parabola which is opening to the right.
  • 7. Describe the difference of x^2/5^2 + y^/3^2 = 1 and x^2/3^2 + y^/5^2 = 1.
  • 8. Describe the difference of x^2/4^2 - y^/3^2 = 1 and x^2/4^2 - y^/3^2 =-1.
  • 9. The equation is (x-2)^2/5^2 + (y+3)^2/3^2 = 1
    • Find the focal length.
    • Find the equation of the principal axis.
    • Find the coordinates of the foci.
    • Find the equation of the directrix.
  • 10 The equation is (x-2)^2/4^2 - (y+3)^2/3^2 = -1
    • Find the focal length.
    • Find the equation of the principal axis.
    • Find the coordinates of the foci.
    • Find the equation of the directrix.
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Q13. Quiz of conic sections

  • 1. See diagram of double cones which we cut in the shapes.
  • 2. (x+2)^2 + (y-2)^2 = 0, hence it is a point.
  • 3. (x+2)^2 + (y-2)^2 = 16 = 4^2, hence it is a circle center at (-2,2) and r = 4.
  • 4. (x+2)^2 + (y-2)^2 = -2, the equation does not exist in real number system.
  • 5. The function is y = x^2 - 6*x + 8.
    • a. xv = -B/(2*A) = 3 and yv = -1 and vertex is at (3,-1).
    • b. Equation of the principal axis is x = 3.
    • c. Since y - k = (x-h)^2/(2*D) and coefficient of x^2 = 1 = 1/(2*D). D=1/2.
    • d. Coordinates of focus is (xv,yv+D/2). That is (3,-0.75).
    • e. Equation of the directrix is y = yv-D/2. That is y = -1.25.
    • f. T openning direction of this parabola is upward.
  • 6. (x-h) = (y-k)^2/(2*D).
  • 7. x^2/5^2 + y^/3^2 = 1
    • Principal axis is y = 0.
    • Foci are ar (-4,0) and (4,0).
    • vertice are at (-5,0) and (5,0).
  • 7. x^2/3^2 + y^/5^2 = 1.
    • Principal axis is x = 0.
    • Foci are ar (0,-4) and (0,4).
    • vertice are at (0,-5) and (0,5).
  • 8. x^2/4^2 - y^/3^2 = 1
    • Principal axis is y = 0.
    • Foci are ar (-5,0) and (5,0).
    • vertice are at (-4,0) and (4,0).
  • 8. x^2/4^2 + y^/3^2 = 1.
    • Principal axis is x = 0.
    • Foci are ar (0,-5) and (0,5).
    • vertice are at (0,-4) and (0,4).
  • 9. The equation is (x-2)^2/5^2 + (y+3)^2/3^2 = 1
    • The focal length f = Sqr(5^2 - 3^2) = 4.
    • The equation of the principal axis is y = -3.
    • The coordinates of the foci are F(2-4,-3) and G(2+4,-3).
    • Find the equation of the directrix.
      • At focus F, the directrix is x = -2 - D.
      • At focus G, the directrix is x = +6 + D.
      • D = (a-f)*(1+e)/e and e = f/a.
      • Hence e = 4/5 = 0.8 and D = (5-4)*(1+0.8)/0.8 = 2.25
  • 10 The equation is (x-2)^2/4^2 - (y+3)^2/3^2 = -1
    • Find the focal length f = (4^2 + 3^2) = 5.
    • Tquation of the principal axis is x = 2.
    • Find the coordinates of the foci are (2,-3-5) and (2,-3+5).
    • Find the equation of the directrix.
      • At focus F, the directrix is x = -2 - D.
      • At focus G, the directrix is x = +6 + D.
      • D = (f-a)*(1+e)/e and e = f/a.
      • Hence e = 5/4 = 1.25 and D = (5-4)*(1+1.25)/1.25 = ?
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