The Central Idea
A conic section is a curve obtained by intersecting a plane with a double cone.
Depending on the position and orientation of the cutting plane, the intersection can be a:
- circle;
- ellipse;
- parabola;
- hyperbola.
These curves can also be defined by distance relationships involving centers, foci, and directrices.
The Double Cone
A double cone consists of two congruent conical halves meeting at a common vertex. Each half is called a nappe.
When a plane intersects this surface, the points shared by the plane and cone form a conic section.
How the Four Main Conics Arise
- A circle occurs when the plane is perpendicular to the cone's axis.
- An ellipse occurs when the plane cuts one nappe obliquely without becoming parallel to a generating line.
- A parabola occurs when the plane is parallel to a generating line of the cone.
- A hyperbola occurs when the plane intersects both nappes.
Degenerate Conics
If the cutting plane passes through the cone's vertex, the intersection can collapse into a simpler set.
Depending on the equation and geometric arrangement, a degenerate conic can be:
- a point;
- a line;
- two intersecting lines;
- two parallel lines;
- or no real points.
Conics as Quadratic Curves
A general second-degree equation in two variables has the form
\[ \boxed{Ax^2+Bxy+Cy^2+Dx+Ey+F=0}. \]Under suitable conditions, its graph is a conic section.
Translation and rotation of coordinates can often transform this equation into one of the standard conic forms.
The Circle
A circle is the set of all points in a plane that are the same distance from a fixed point called the center.
If the center is \((h,k)\) and the radius is \(r>0\), then
\[ \sqrt{(x-h)^2+(y-k)^2}=r. \]Squaring gives
\[ \boxed{(x-h)^2+(y-k)^2=r^2}. \]Example: Reading a Circle
Consider
\[ (x-3)^2+(y+2)^2=25. \]The center is
\[ \boxed{(3,-2)} \]and the radius is
\[ \boxed{5}. \]Circle Centered at the Origin
When the center is \((0,0)\), the equation becomes
\[ \boxed{x^2+y^2=r^2}. \]The unit circle is
\[ \boxed{x^2+y^2=1}. \]Equal coefficients on \(x^2\) and \(y^2\) reflect the circle's complete rotational symmetry.
The Ellipse
An ellipse is the set of all points for which the sum of the distances to two fixed points is constant.
The fixed points are the foci.
If \(P\) is on the ellipse and the foci are \(F_1\) and \(F_2\), then
\[ \boxed{PF_1+PF_2=\text{constant}}. \]A circle is the special case in which the two foci coincide.
Standard Horizontal Ellipse
An ellipse centered at the origin with horizontal major axis has equation
\[ \boxed{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}, \qquad a>b>0. \]The vertices are
\[ (\pm a,0), \]and the co-vertices are
\[ (0,\pm b). \]Ellipse Foci
For a horizontal ellipse,
\[ \boxed{c^2=a^2-b^2}. \]The foci are
\[ \boxed{(\pm c,0)}. \]
Since \(c
Standard Vertical Ellipse
If the major axis is vertical,
\[ \boxed{\frac{x^2}{b^2}+\frac{y^2}{a^2}=1}, \qquad a>b>0. \]The vertices are
\[ (0,\pm a), \]and the foci are
\[ \boxed{(0,\pm c)}. \]Translated Ellipse
A horizontal ellipse centered at \((h,k)\) has equation
\[ \boxed{\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1}, \qquad a>b>0. \]Its foci are
\[ \boxed{(h\pm c,k)}. \]Example: Reading an Ellipse
Consider
\[ \frac{(x-2)^2}{25}+\frac{(y+1)^2}{9}=1. \]The center is
\[ (2,-1). \]Since
\[ a=5,\qquad b=3, \]the major axis is horizontal.
The focal distance is
\[ c=\sqrt{25-9}=4. \]Therefore, the foci are
\[ \boxed{(-2,-1)\quad\text{and}\quad(6,-1)}. \]The Parabola
A parabola is the set of all points equidistant from:
- a fixed point called the focus;
- a fixed line called the directrix.
If \(P\) is on the parabola and \(\ell\) is the directrix, then
\[ \boxed{PF=d(P,\ell)}. \]Vertex and Axis of a Parabola
The vertex lies halfway between the focus and directrix along the axis of symmetry.
The axis passes through the vertex and focus and is perpendicular to the directrix.
Standard Vertical Parabola
A vertical parabola with vertex \((h,k)\) has equation
\[ \boxed{(x-h)^2=4p(y-k)}. \]Its focus is
\[ \boxed{(h,k+p)}, \]and its directrix is
\[ \boxed{y=k-p}. \]If \(p>0\), it opens upward. If \(p<0\), it opens downward.
Standard Horizontal Parabola
A horizontal parabola has equation
\[ \boxed{(y-k)^2=4p(x-h)}. \]Its focus is
\[ \boxed{(h+p,k)}, \]and its directrix is
\[ \boxed{x=h-p}. \]If \(p>0\), it opens right. If \(p<0\), it opens left.
Example: Reading a Parabola
Consider
\[ (x-1)^2=12(y+2). \]Since
\[ 4p=12, \]we have
\[ p=3. \]The vertex is
\[ \boxed{(1,-2)}, \]the focus is
\[ \boxed{(1,1)}, \]and the directrix is
\[ \boxed{y=-5}. \]The Hyperbola
A hyperbola is the set of all points for which the absolute value of the difference of the distances to two fixed points is constant.
The fixed points are the foci.
\[ \boxed{|PF_1-PF_2|=\text{constant}}. \]A nondegenerate hyperbola has two separate branches.
Standard Horizontal Hyperbola
A horizontal hyperbola centered at the origin has equation
\[ \boxed{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}. \]The vertices are
\[ (\pm a,0), \]and the foci are
\[ (\pm c,0), \]where
\[ \boxed{c^2=a^2+b^2}. \]Standard Vertical Hyperbola
A vertical hyperbola centered at the origin has equation
\[ \boxed{\frac{y^2}{a^2}-\frac{x^2}{b^2}=1}. \]The vertices are
\[ (0,\pm a), \]and the foci are
\[ \boxed{(0,\pm c)}. \]Translated Hyperbola
A horizontal hyperbola centered at \((h,k)\) has equation
\[ \boxed{\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1}. \]A vertical hyperbola has equation
\[ \boxed{\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1}. \]Hyperbola Asymptotes
For
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \]the asymptotes are
\[ \boxed{y=\pm\frac{b}{a}x}. \]For
\[ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1, \]they become
\[ \boxed{y-k=\pm\frac{b}{a}(x-h)}. \]Example: Reading a Hyperbola
Consider
\[ \frac{(x+1)^2}{16}-\frac{(y-2)^2}{9}=1. \]The center is \((-1,2)\), with
\[ a=4,\qquad b=3. \]Since
\[ c^2=16+9=25, \]we have
\[ c=5. \]The foci are
\[ \boxed{(-6,2)\quad\text{and}\quad(4,2)}. \]Ellipse and Hyperbola: The Key Sign Difference
Compare
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \]with
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1. \]The ellipse uses a sum of positive squared terms.
The hyperbola uses a difference.
That sign change corresponds to a geometric change from a bounded closed curve to two unbounded branches.
Ellipse and Hyperbola: The Focal Formulas
For an ellipse,
\[ \boxed{c^2=a^2-b^2}. \]For a hyperbola,
\[ \boxed{c^2=a^2+b^2}. \]Thus ellipse foci lie between the center and vertices, while hyperbola foci lie beyond the vertices.
Eccentricity
Eccentricity, denoted by \(e\), gives a unified numerical classification of conics.
- circle: \(e=0\);
- ellipse: \(0<e<1\);
- parabola: \(e=1\);
- hyperbola: \(e>1\).
Eccentricity of an Ellipse
For an ellipse,
\[ \boxed{e=\frac{c}{a}}. \]
Since \(c
Eccentricity of a Hyperbola
For a hyperbola,
\[ \boxed{e=\frac{c}{a}}. \]Since \(c>a\),
\[ \boxed{e>1}. \]The Focus-Directrix Definition
A broad definition of a conic uses one focus, one directrix, and eccentricity.
A point \(P\) lies on the conic when
\[ \boxed{\frac{PF}{d(P,\ell)}=e}. \]The value of \(e\) determines whether the curve is ellipse-type, parabolic, or hyperbolic.
Why a Parabola Has Eccentricity \(1\)
A parabola satisfies
\[ PF=d(P,\ell). \]Therefore,
\[ \frac{PF}{d(P,\ell)}=1, \]so
\[ \boxed{e=1}. \]The Circle as the Symmetric Ellipse Case
If an ellipse has
\[ a=b, \]then
\[ c^2=a^2-b^2=0. \]Thus
\[ c=0 \]and the two foci coincide at the center.
Its eccentricity becomes
\[ \boxed{e=0}. \]Completing the Square to Reveal a Conic
Expanded quadratic equations often hide the geometric form.
Consider
\[ x^2+y^2-6x+4y-12=0. \]Group:
\[ (x^2-6x)+(y^2+4y)=12. \]Complete the squares:
\[ (x-3)^2-9+(y+2)^2-4=12. \]Therefore,
\[ \boxed{(x-3)^2+(y+2)^2=25}. \]Example: Completing the Square for a Parabola
Consider
\[ x^2-4x-8y+12=0. \]Rearrange:
\[ x^2-4x=8y-12. \]Complete the square:
\[ x^2-4x+4=8y-8. \]Then
\[ \boxed{(x-2)^2=8(y-1)}. \]The vertex is \((2,1)\) and \(p=2\).
Example: Completing the Square for an Ellipse
Consider
\[ 4x^2+9y^2-8x+36y+4=0. \]Group:
\[ 4(x^2-2x)+9(y^2+4y)+4=0. \]Complete the squares:
\[ 4[(x-1)^2-1]+9[(y+2)^2-4]+4=0. \]Simplify:
\[ 4(x-1)^2+9(y+2)^2=36. \]Therefore,
\[ \boxed{\frac{(x-1)^2}{9}+\frac{(y+2)^2}{4}=1}. \]Example: Completing the Square for a Hyperbola
Consider
\[ 9x^2-4y^2-18x-16y-43=0. \]Group:
\[ 9(x^2-2x)-4(y^2+4y)=43. \]Complete the squares:
\[ 9[(x-1)^2-1]-4[(y+2)^2-4]=43. \]Simplify:
\[ 9(x-1)^2-4(y+2)^2=36. \]Therefore,
\[ \boxed{\frac{(x-1)^2}{4}-\frac{(y+2)^2}{9}=1}. \]Recognizing Conics from Squared Terms
When there is no \(xy\)-term and the equation has been simplified:
- same-sign squared terms with equal coefficients suggest a circle;
- same-sign squared terms with unequal coefficients suggest an ellipse;
- one squared variable suggests a parabola;
- opposite-sign squared terms suggest a hyperbola.
Degenerate cases still require separate checking.
The Discriminant \(B^2-4AC\)
For
\[ Ax^2+Bxy+Cy^2+Dx+Ey+F=0, \]the quantity
\[ \boxed{B^2-4AC} \]helps classify the quadratic curve.
For a nondegenerate real conic:
- \(B^2-4AC<0\): ellipse-type, including circles;
- \(B^2-4AC=0\): parabola-type;
- \(B^2-4AC>0\): hyperbola-type.
Why the \(xy\)-Term Matters
A mixed term
\[ Bxy \]often indicates that the conic's natural axes are rotated relative to the coordinate axes.
A familiar conic can therefore look unfamiliar simply because it is tilted.
Rotation of Axes
A rotated coordinate system can be introduced by
\[ x=X\cos\phi-Y\sin\phi, \] \[ y=X\sin\phi+Y\cos\phi. \]For a suitable angle \(\phi\), the mixed \(XY\)-term can often be removed.
The important idea here is geometric: changing coordinates can simplify the equation without changing the curve itself.
Conics in Parametric Form
Conics can also be represented parametrically.
A circle of radius \(r\) centered at the origin can be written
\[ \boxed{x=r\cos t,\qquad y=r\sin t}. \]An ellipse can be written
\[ \boxed{x=a\cos t,\qquad y=b\sin t}. \]Parametric form makes direction and traversal explicit.
Parametric Parabola
The parabola
\[ y=x^2 \]has the simple parametrization
\[ \boxed{x=t,\qquad y=t^2}. \]Parametric Hyperbola
The identity
\[ \sec^2 t-\tan^2 t=1 \]gives a parametrization of
\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1: \] \[ \boxed{x=a\sec t,\qquad y=b\tan t}, \]wherever the trigonometric functions are defined.
Conics in Polar Form
When a focus is placed at the pole, a conic can be represented in a unified polar form such as
\[ \boxed{r=\frac{\ell}{1+e\cos\theta}}, \]where \(e\) is eccentricity and \(\ell>0\) is a scale parameter.
Related orientations use \(\sin\theta\) or a different sign.
Polar Classification by Eccentricity
In the polar conic form,
- \(0\le e<1\): ellipse-type, including \(e=0\) for a circle;
- \(e=1\): parabola;
- \(e>1\): hyperbola.
This is one of the clearest ways to see the four conics as members of a single family.
Reflection Property of a Parabola
Under the ideal geometric reflection law, rays traveling parallel to a parabola's axis reflect toward the focus.
Conversely, rays originating at the focus reflect outward parallel to the axis.
This property underlies idealized parabolic reflectors, antennas, telescopes, and headlights.
Reflection Property of an Ellipse
Under the same ideal reflection law, a ray leaving one focus of an ellipse and reflecting from the ellipse travels toward the other focus.
This property is connected directly to the ellipse's constant-sum distance definition.
Conics and Symmetry
- A circle has infinitely many reflection axes through its center.
- An ellipse has two principal reflection axes.
- A parabola has one reflection axis.
- A hyperbola has two principal reflection axes and \(180^\circ\) rotational symmetry about its center.
Conics and Boundedness
- Circle: bounded and closed.
- Ellipse: bounded and closed.
- Parabola: unbounded and open.
- Hyperbola: unbounded and divided into two branches.
Conics as Level Curves
Conics can be interpreted as level curves of quadratic expressions.
For example,
\[ x^2+y^2=9 \]is a level curve of
\[ F(x,y)=x^2+y^2. \]This viewpoint connects conic geometry with later work on level curves and level surfaces.
From Conics to Quadratic Surfaces
The algebraic patterns of conics extend naturally into three dimensions.
For example,
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \]describes an ellipse, while
\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \]describes an ellipsoid.
Cross-Sections of Surfaces
Conics frequently appear as planar cross-sections of three-dimensional surfaces.
A sphere produces circular sections. An ellipsoid produces elliptical sections. Other quadratic surfaces can produce ellipses, parabolas, or hyperbolas depending on the cutting plane.
Common Error: Confusing \(a\), \(b\), and \(c\)
For an ellipse,
\[ c^2=a^2-b^2. \]For a hyperbola,
\[ c^2=a^2+b^2. \]These formulas should not be interchanged.
Common Error: Assuming the Larger Denominator Always Lies Under \(x^2\)
For an ellipse, the larger denominator identifies the major-axis direction.
Thus
\[ \frac{x^2}{25}+\frac{y^2}{9}=1 \]has a horizontal major axis, while
\[ \frac{x^2}{9}+\frac{y^2}{25}=1 \]has a vertical major axis.
Common Error: Using Denominator Size to Determine Hyperbola Direction
For a hyperbola, the opening direction is determined by the positive squared term.
For
\[ \frac{x^2}{4}-\frac{y^2}{100}=1, \]the hyperbola opens left and right even though \(100>4\).
Common Error: Forgetting the Factor \(4p\)
In
\[ (x-h)^2=4p(y-k), \]the coefficient is \(4p\), not \(p\).
If
\[ (x-h)^2=20(y-k), \]then
\[ p=5. \]Common Error: Reading Translation Signs Incorrectly
The equation
\[ (x+3)^2+(y-5)^2=16 \]has center
\[ \boxed{(-3,5)}, \]not \((3,-5)\).
Common Error: Ignoring Degenerate Cases
The equation
\[ x^2-y^2=0 \]factors as
\[ (x-y)(x+y)=0. \]It represents two intersecting lines, not a nondegenerate hyperbola.
Common Error: Assuming Every Quadratic Equation Has Real Points
Consider
\[ x^2+y^2=-1. \]Since
\[ x^2+y^2\ge 0 \]for real \(x\) and \(y\), this equation has no real points.
A Reliable Procedure for Identifying a Conic
- Move terms into a convenient arrangement.
- Group \(x\)-terms and \(y\)-terms.
- Complete squares when linear terms are present.
- Normalize the equation when appropriate.
- Inspect the number and signs of squared terms.
- Identify the center or vertex.
- Read the principal lengths or \(4p\).
- Compute focal data if required.
- Check for degeneracy and real feasibility.
Example: Full Identification Workflow
Identify
\[ 9x^2+4y^2-54x+8y+49=0. \]Group:
\[ 9(x^2-6x)+4(y^2+2y)+49=0. \]Complete the squares:
\[ 9[(x-3)^2-9]+4[(y+1)^2-1]+49=0. \]Simplify:
\[ 9(x-3)^2+4(y+1)^2=36. \]Divide by \(36\):
\[ \boxed{\frac{(x-3)^2}{4}+\frac{(y+1)^2}{9}=1}. \]This is an ellipse centered at
\[ (3,-1) \]with vertical major axis.
Here
\[ a=3,\qquad b=2,\qquad c=\sqrt5. \]The foci are
\[ \boxed{(3,-1\pm\sqrt5)}. \]Practice 1: Circle
Find the center and radius of
\[ (x+2)^2+(y-4)^2=36. \]Practice 2: Ellipse
For
\[ \frac{x^2}{36}+\frac{y^2}{20}=1, \]find \(a\), \(b\), \(c\), and the foci.
Practice 3: Parabola
For
\[ (y+1)^2=16(x-3), \]find the vertex, focus, directrix, and opening direction.
Practice 4: Hyperbola
For
\[ \frac{(y-2)^2}{25}-\frac{(x+1)^2}{9}=1, \]find the center, vertices, and foci.
Practice 5: Identify the Conic
Classify
\[ 4x^2+9y^2=36. \]Practice 6: Complete the Square
Rewrite in standard form and identify:
\[ x^2+y^2-8x+6y-11=0. \]Practice 7: Complete the Square
Rewrite in standard form and identify:
\[ y^2-6y-8x+1=0. \]Practice 8: Discriminant Classification
Use \(B^2-4AC\) to determine the conic type suggested by
\[ 3x^2+4xy+3y^2-1=0. \]Practice 9: Eccentricity
An ellipse has
\[ a=10,\qquad c=6. \]Find its eccentricity.
Practice 10: Degenerate Conic
Classify the real point set represented by
\[ x^2-4y^2=0. \]Practice Answers
Practice 1
The center is
\[ \boxed{(-2,4)} \]and the radius is
\[ \boxed{6}. \]Practice 2
We have
\[ a=6,\qquad b=2\sqrt5. \]Then
\[ c^2=36-20=16, \]so
\[ c=4. \]The foci are
\[ \boxed{(\pm4,0)}. \]Practice 3
Here
\[ h=3,\qquad k=-1,\qquad 4p=16, \]so
\[ p=4. \]The vertex is
\[ \boxed{(3,-1)}, \]the focus is
\[ \boxed{(7,-1)}, \]the directrix is
\[ \boxed{x=-1}, \]and the parabola opens right.
Practice 4
The center is
\[ \boxed{(-1,2)}. \]Since the \(y\)-term is positive, the hyperbola opens vertically.
We have
\[ a=5,\qquad b=3. \]Thus
\[ c=\sqrt{25+9}=\sqrt{34}. \]The vertices are
\[ \boxed{(-1,7)\quad\text{and}\quad(-1,-3)}, \]and the foci are
\[ \boxed{(-1,2\pm\sqrt{34})}. \]Practice 5
Divide by \(36\):
\[ \frac{x^2}{9}+\frac{y^2}{4}=1. \]Therefore, the conic is an
\[ \boxed{\text{ellipse}}. \]Practice 6
Complete the squares:
\[ (x-4)^2+(y+3)^2=36. \]Therefore, the conic is a circle centered at \((4,-3)\) with radius \(6\).
Practice 7
Rearrange and complete the square:
\[ y^2-6y=8x-1, \] \[ y^2-6y+9=8x+8. \]Therefore,
\[ \boxed{(y-3)^2=8(x+1)}. \]The conic is a parabola.
Practice 8
Here
\[ A=3,\qquad B=4,\qquad C=3. \]Therefore,
\[ B^2-4AC=16-36=-20<0. \]The equation is
\[ \boxed{\text{ellipse-type}}. \]Practice 9
Use
\[ e=\frac{c}{a}. \]Thus
\[ \boxed{e=\frac{6}{10}=\frac35}. \]Practice 10
Factor:
\[ x^2-4y^2=(x-2y)(x+2y). \]Therefore,
\[ \boxed{x=2y\quad\text{or}\quad x=-2y}. \]The equation represents two intersecting lines, a degenerate conic.
What to Remember
The four principal conics form one geometric family.
- Circle: constant distance from a center, \[ (x-h)^2+(y-k)^2=r^2. \]
- Ellipse: constant sum of distances to two foci, \[ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1. \]
- Parabola: equal distance from a focus and directrix, \[ (x-h)^2=4p(y-k) \] or its horizontal counterpart.
- Hyperbola: constant absolute difference of distances to two foci, \[ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \] or its vertical counterpart.
Eccentricity unifies the classification:
\[ \boxed{ e=0\text{ circle},\quad 0Where We Go Next
Conic sections arise from a particularly simple geometric operation: slicing a cone with a plane. Many other curves arise through different generating processes.
The next lesson, Spirals, Cycloids, and Generated Curves, studies curves created by rotation, rolling motion, changing radius, and other geometric mechanisms.