The Central Idea
A conic section is a curve that can be described both geometrically and algebraically.
The four classical conics are:
- circles;
- ellipses;
- parabolas;
- hyperbolas.
Each can be represented by equations in \(x\) and \(y\), and each can be understood through a characteristic distance relationship.
Why They Are Called Conic Sections
The classical conics can be produced by intersecting a plane with a double cone.
Different orientations of the cutting plane produce different curves.
Coordinate geometry allows us to study the same curves without requiring the three-dimensional cone construction every time.
Circles as Distance Loci
A circle is the set of all points at a fixed distance \(r\) from a fixed point called the center.
If the center is
\[ (h,k) \]and a point on the circle is
\[ (x,y), \]then the distance formula gives
\[ \sqrt{(x-h)^2+(y-k)^2}=r. \]Standard Equation of a Circle
Squaring the distance equation gives
\[ \boxed{(x-h)^2+(y-k)^2=r^2}. \]This is the standard equation of a circle with center
\[ \boxed{(h,k)} \]and radius
\[ \boxed{r}. \]Circle Centered at the Origin
If
\[ h=0,\qquad k=0, \]then the circle equation becomes
\[ \boxed{x^2+y^2=r^2}. \]For example,
\[ x^2+y^2=25 \]has center \((0,0)\) and radius \(5\).
Reading a Circle Equation
Consider
\[ (x-3)^2+(y+4)^2=49. \]Compare with
\[ (x-h)^2+(y-k)^2=r^2. \]Therefore,
\[ h=3,\qquad k=-4,\qquad r=7. \]The center is
\[ \boxed{(3,-4)} \]and the radius is
\[ \boxed{7}. \]Why the Center Signs Look Reversed
In
\[ (x-h)^2+(y-k)^2=r^2, \]the center coordinates are subtracted inside the parentheses.
Therefore
\[ (x+5)^2 \]means
\[ x-(-5), \]so the horizontal center coordinate is \(-5\).
Constructing a Circle Equation
Write the equation of a circle with center
\[ (-2,6) \]and radius
\[ 4. \]Substitute into standard form:
\[ \boxed{(x+2)^2+(y-6)^2=16}. \]Diameter and Radius
If the endpoints of a diameter are known, the center is their midpoint.
Suppose a diameter has endpoints
\[ A=(-1,2) \]and
\[ B=(7,8). \]The center is
\[ \left( \frac{-1+7}{2}, \frac{2+8}{2} \right) = (3,5). \]The radius is half the distance \(AB\).
Example: Circle from Diameter Endpoints
Continue with
\[ A=(-1,2),\qquad B=(7,8). \]The diameter length is
\[ AB= \sqrt{(7-(-1))^2+(8-2)^2} = \sqrt{8^2+6^2} = 10. \]Therefore the radius is
\[ r=5. \]The circle equation is
\[ \boxed{(x-3)^2+(y-5)^2=25}. \]General Form of a Circle
Expanding standard form produces an equation of the form
\[ \boxed{x^2+y^2+Dx+Ey+F=0}. \]A circle in this form has equal coefficients on \(x^2\) and \(y^2\) and no \(xy\)-term after suitable normalization.
Completing the Square for a Circle
Consider
\[ x^2+y^2-6x+8y-11=0. \]Group the \(x\)- and \(y\)-terms:
\[ (x^2-6x)+(y^2+8y)=11. \]Complete each square:
\[ x^2-6x+9=(x-3)^2, \] \[ y^2+8y+16=(y+4)^2. \]Add the same values to both sides:
\[ (x-3)^2+(y+4)^2=36. \]Therefore the center is \((3,-4)\) and the radius is \(6\).
Conics as Quadratic Curves
Circles, ellipses, parabolas, and hyperbolas are all described by second-degree equations.
A general quadratic equation in two variables is
\[ \boxed{ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 }. \]Different coefficient patterns produce different conic types.
Ellipses
An ellipse is the set of points for which the sum of the distances to two fixed points, called foci, is constant.
A circle is a special ellipse in which the two foci coincide at the center.
Standard Horizontal Ellipse
An ellipse centered at
\[ (h,k) \]with horizontal major axis has equation
\[ \boxed{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1 }, \]where
\[ a>b>0. \]The larger denominator identifies the major-axis direction.
Standard Vertical Ellipse
A vertically oriented ellipse centered at
\[ (h,k) \]has equation
\[ \boxed{ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} =1 }, \]where
\[ a>b>0. \]Ellipse Vertices and Co-Vertices
For a horizontal ellipse,
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1, \]the vertices are
\[ \boxed{(h\pm a,k)}, \]and the co-vertices are
\[ \boxed{(h,k\pm b)}. \]Ellipse Foci
For an ellipse,
\[ \boxed{c^2=a^2-b^2}. \]For a horizontal major axis, the foci are
\[ \boxed{(h\pm c,k)}. \]For a vertical major axis, the foci are
\[ \boxed{(h,k\pm c)}. \]Example: Reading an Ellipse
Consider
\[ \frac{(x-2)^2}{25} + \frac{(y+1)^2}{9} =1. \]The center is
\[ (2,-1). \]Since
\[ a^2=25,\qquad b^2=9, \]the major axis is horizontal and
\[ a=5,\qquad b=3. \]Also,
\[ c^2=25-9=16, \]so
\[ c=4. \]Circle as a Special Ellipse
If
\[ a=b=r, \]then
\[ \frac{(x-h)^2}{r^2} + \frac{(y-k)^2}{r^2} =1. \]Multiply by \(r^2\):
\[ (x-h)^2+(y-k)^2=r^2. \]This is precisely the circle equation.
Parabolas
A parabola is the set of all points equidistant from:
- a fixed point called the focus;
- a fixed line called the directrix.
The midpoint between the focus and directrix along the axis of symmetry is the vertex.
Vertical Parabola
A parabola with vertex
\[ (h,k) \]and vertical axis has standard equation
\[ \boxed{(x-h)^2=4p(y-k)}. \]Its focus is
\[ \boxed{(h,k+p)}, \]and its directrix is
\[ \boxed{y=k-p}. \]Horizontal Parabola
A parabola with horizontal axis has equation
\[ \boxed{(y-k)^2=4p(x-h)}. \]Its focus is
\[ \boxed{(h+p,k)}, \]and its directrix is
\[ \boxed{x=h-p}. \]The Sign of \(p\)
The parameter \(p\) determines the opening direction.
- For \((x-h)^2=4p(y-k)\), \(p>0\) opens upward and \(p<0\) opens downward.
- For \((y-k)^2=4p(x-h)\), \(p>0\) opens right and \(p<0\) opens left.
Example: Reading a Parabola
Consider
\[ (x-3)^2=8(y+2). \]The vertex is
\[ (3,-2). \]Since
\[ 4p=8, \]we have
\[ p=2. \]Therefore the focus is
\[ \boxed{(3,0)}, \]and the directrix is
\[ \boxed{y=-4}. \]Parabolas and Quadratic Functions
A vertical parabola can also be written as
\[ y=a(x-h)^2+k. \]This is the familiar vertex form of a quadratic function.
Comparing with
\[ (x-h)^2=4p(y-k), \]gives
\[ a=\frac{1}{4p}. \]The geometric focus-directrix description and the algebraic quadratic function are therefore two views of the same curve.
Hyperbolas
A hyperbola is the set of points for which the absolute difference of the distances to two fixed foci is constant.
Unlike an ellipse, a hyperbola has two separate branches.
Horizontal Hyperbola
A hyperbola centered at
\[ (h,k) \]with horizontal transverse axis has equation
\[ \boxed{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1 }. \]The branches open left and right.
Vertical Hyperbola
A hyperbola with vertical transverse axis has equation
\[ \boxed{ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} =1 }. \]The branches open upward and downward.
Hyperbola Vertices and Foci
For a horizontal hyperbola,
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1, \]the vertices are
\[ \boxed{(h\pm a,k)}. \]The foci satisfy
\[ \boxed{c^2=a^2+b^2}, \]so they are
\[ \boxed{(h\pm c,k)}. \]Hyperbola Asymptotes
For the horizontal hyperbola
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1, \]the asymptotes are
\[ \boxed{ y-k=\pm\frac{b}{a}(x-h) }. \]For a vertical hyperbola,
\[ \boxed{ y-k=\pm\frac{a}{b}(x-h) }. \]Example: Reading a Hyperbola
Consider
\[ \frac{(x+1)^2}{16} - \frac{(y-2)^2}{9} =1. \]The center is
\[ (-1,2). \]Since the \(x\)-term is positive, the hyperbola opens left and right.
We have
\[ a=4,\qquad b=3. \]Therefore,
\[ c^2=16+9=25, \]so
\[ c=5. \]Comparing Ellipses and Hyperbolas
The standard equations look similar but differ in a critical sign.
Ellipse:
\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} =1. \]Hyperbola:
\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1. \]The plus sign keeps the ellipse bounded. The subtraction produces two unbounded hyperbola branches.
Eccentricity
Eccentricity provides a unified measure of how a conic departs from circularity.
It is commonly denoted by
\[ e. \]For the classical conics:
- circle: \(e=0\);
- ellipse: \(0<e<1\);
- parabola: \(e=1\);
- hyperbola: \(e>1\).
Eccentricity of an Ellipse
For an ellipse,
\[ \boxed{e=\frac{c}{a}}, \]where
\[ c^2=a^2-b^2. \]A smaller eccentricity gives a more circle-like ellipse, while an eccentricity closer to \(1\) gives a more elongated ellipse.
Eccentricity of a Hyperbola
For a hyperbola,
\[ \boxed{e=\frac{c}{a}}, \]where
\[ c^2=a^2+b^2. \]Since \(c>a\), hyperbola eccentricity is always greater than \(1\).
Translation of Conics
Moving a conic away from the origin changes its equations by replacing
\[ x \]with
\[ x-h \]and
\[ y \]with
\[ y-k. \]This is the same coordinate-translation structure studied in the previous lesson.
Symmetry of Conics
Standard conics have characteristic symmetries.
- A circle is symmetric about every diameter through its center.
- An ellipse is symmetric about both principal axes through its center.
- A parabola is symmetric about one axis.
- A hyperbola is symmetric about both principal axes and through its center.
Coordinate equations make these symmetries visible through squared terms.
Degenerate Conics
Some quadratic equations produce limiting or collapsed cases rather than ordinary conic curves.
Examples can include:
- a single point;
- a pair of intersecting lines;
- a pair of parallel lines;
- an empty real solution set.
These are called degenerate conics.
Example: A Degenerate Circle Equation
Consider
\[ x^2+y^2=0. \]Since squares are nonnegative, both must be zero:
\[ x=0,\qquad y=0. \]Therefore the entire solution set is the single point
\[ \boxed{(0,0)}. \]This may be viewed as a circle of radius zero.
Identifying Conics from Standard Forms
A quick structural comparison is often enough.
- Two squared terms added with equal denominators: circle.
- Two squared terms added with unequal denominators: ellipse.
- Only one variable squared: parabola.
- Two squared terms with opposite signs: hyperbola.
These rules apply most directly after the equation has been rewritten into standard form.
The Discriminant Classification
For the general quadratic
\[ Ax^2+Bxy+Cy^2+Dx+Ey+F=0, \]the quantity
\[ \boxed{B^2-4AC} \]helps classify the conic, assuming a nondegenerate real curve.
- If \(B^2-4AC<0\), the conic is ellipse-type.
- If \(B^2-4AC=0\), the conic is parabola-type.
- If \(B^2-4AC>0\), the conic is hyperbola-type.
A circle is a special ellipse-type case.
Rotated Conics
If an equation contains an
\[ xy \]term, the conic's principal axes may be rotated relative to the coordinate axes.
A rotation of coordinates can remove the cross term in appropriate cases, revealing a standard conic equation.
This is a deeper application of coordinate transformations.
Conics as Loci
Each conic is fundamentally a locus defined by a geometric condition.
- Circle: fixed distance from one center.
- Ellipse: fixed sum of distances to two foci.
- Parabola: equal distance from a focus and directrix.
- Hyperbola: fixed absolute difference of distances to two foci.
These definitions explain why the equations have the forms they do.
Conics in Applications
Conic sections appear throughout mathematics and applied science.
- Circles model radial distance and rotation.
- Ellipses appear in orbital models and reflective systems.
- Parabolas appear in projectile models and focusing reflectors.
- Hyperbolas appear in navigation, signal timing, and inverse relationships.
Their importance comes from both their geometric simplicity and their algebraic structure.
Common Error: Reading the Center Signs Incorrectly
In
\[ (x+2)^2+(y-5)^2=16, \]the center is
\[ \boxed{(-2,5)}, \]not \((2,-5)\).
Always compare with
\[ (x-h)^2+(y-k)^2=r^2. \]Common Error: Using \(r^2\) as the Radius
If
\[ (x-h)^2+(y-k)^2=49, \]then
\[ r^2=49. \]Therefore,
\[ \boxed{r=7}, \]not \(49\).
Common Error: Confusing \(a\), \(b\), and Their Squares
In
\[ \frac{x^2}{25}+\frac{y^2}{9}=1, \]the denominators are
\[ a^2=25,\qquad b^2=9. \]Therefore,
\[ a=5,\qquad b=3. \]The semi-axis lengths are the square roots of the denominators.
Common Error: Using the Ellipse Focus Formula for a Hyperbola
For an ellipse,
\[ c^2=a^2-b^2. \]For a hyperbola,
\[ c^2=a^2+b^2. \]The sign difference is essential.
Common Error: Choosing the Wrong Parabola Orientation
In
\[ (x-h)^2=4p(y-k), \]the \(x\)-expression is squared, so the parabola opens vertically.
In
\[ (y-k)^2=4p(x-h), \]the \(y\)-expression is squared, so the parabola opens horizontally.
A Reliable Procedure for Reading a Conic Equation
- Rewrite the equation into standard form if necessary.
- Identify which variables are squared.
- Inspect whether the squared terms are added or subtracted.
- Read the center or vertex from translated coordinate terms.
- Read the relevant squared parameters from the denominators or coefficients.
- Determine orientation.
- Compute foci, vertices, asymptotes, or directrices as appropriate.
- Check the geometric interpretation against the equation.
Practice 1: Circle Center and Radius
Find the center and radius of
\[ (x-4)^2+(y+3)^2=36. \]Practice 2: Circle Equation
Write the equation of the circle with center
\[ (-1,5) \]and radius
\[ 3. \]Practice 3: Complete the Square
Rewrite
\[ x^2+y^2+4x-6y-12=0 \]in standard circle form.
Practice 4: Ellipse
For
\[ \frac{x^2}{36}+\frac{y^2}{16}=1, \]identify \(a\), \(b\), and the orientation of the major axis.
Practice 5: Ellipse Foci
Find the foci of
\[ \frac{x^2}{25}+\frac{y^2}{9}=1. \]Practice 6: Parabola
For
\[ (x+2)^2=-12(y-1), \]find the vertex, focus, and directrix.
Practice 7: Hyperbola
For
\[ \frac{x^2}{9}-\frac{y^2}{16}=1, \]identify the center, vertices, and opening direction.
Practice 8: Hyperbola Foci
Find the foci of
\[ \frac{x^2}{16}-\frac{y^2}{9}=1. \]Practice 9: Classify the Conic
Classify
\[ 9x^2+4y^2=36. \]Practice 10: Identify the Error
A student says the circle
\[ (x+6)^2+(y-2)^2=81 \]has center \((6,-2)\) and radius \(81\).
Explain both errors.
Practice Answers
Practice 1
Compare with
\[ (x-h)^2+(y-k)^2=r^2. \]Therefore,
\[ \boxed{\text{center }(4,-3),\quad r=6}. \]Practice 2
\[ \boxed{(x+1)^2+(y-5)^2=9}. \]Practice 3
Group and complete the squares:
\[ (x^2+4x)+(y^2-6y)=12. \] \[ (x+2)^2+(y-3)^2=25. \]Therefore,
\[ \boxed{(x+2)^2+(y-3)^2=25}. \]Practice 4
\[ a^2=36,\qquad b^2=16. \]Therefore,
\[ \boxed{a=6,\qquad b=4}. \]The major axis is horizontal.
Practice 5
\[ c^2=25-9=16, \] \[ c=4. \]The foci are
\[ \boxed{(-4,0)\text{ and }(4,0)}. \]Practice 6
The vertex is
\[ (-2,1). \]Since
\[ 4p=-12, \]we have
\[ p=-3. \]Therefore the focus is
\[ \boxed{(-2,-2)}, \]and the directrix is
\[ \boxed{y=4}. \]Practice 7
The center is
\[ (0,0). \]Since
\[ a=3, \]the vertices are
\[ \boxed{(-3,0)\text{ and }(3,0)}. \]The hyperbola opens left and right.
Practice 8
\[ c^2=16+9=25, \] \[ c=5. \]Therefore the foci are
\[ \boxed{(-5,0)\text{ and }(5,0)}. \]Practice 9
Divide by \(36\):
\[ \frac{x^2}{4}+\frac{y^2}{9}=1. \]This is an
\[ \boxed{\text{ellipse}}. \]Practice 10
The center signs were read incorrectly:
\[ \boxed{(-6,2)}. \]Also,
\[ r^2=81, \]so
\[ \boxed{r=9}. \]What to Remember
The standard circle equation is
\[ \boxed{(x-h)^2+(y-k)^2=r^2}. \]The standard conic forms are distinguished by their squared-variable structure:
- circle: equal squared terms added;
- ellipse: unequal squared terms added;
- parabola: one variable squared;
- hyperbola: squared terms subtracted.
The geometric definitions are equally important:
- circle: fixed distance from one center;
- ellipse: fixed sum of distances to two foci;
- parabola: equal distance from focus and directrix;
- hyperbola: fixed absolute difference of distances to two foci.
Coordinate transformations and completing the square help move between general equations and recognizable standard forms.
Where We Go Next
Conic equations describe curves implicitly or as relations between coordinates. Another powerful approach is to describe both coordinates in terms of a third variable that traces the curve.
The next lesson, Parametric Equations, develops this method and shows how points can move along lines, circles, and more general curves as a parameter changes.