Unit 10 is about the four curves you get by slicing a double cone with a plane: circles, parabolas, ellipses and hyperbolas. By the end, you should be able to write and graph each one in standard form, find its center or focus and, for an ellipse or hyperbola, its foci, and solve systems where these curves intersect.
Read the overview first, then work through each topic's notes and try the check-yourself question before peeking at the answer.
Conic sections show up when a plane cuts through a double cone at different angles. In algebra, each one is a second-degree equation in and . This unit teaches you to recognize which conic an equation describes, rewrite it in a form you can graph, and find where curves meet.
Honors Algebra II Unit 10 covers conic sections. You learn the distance-based definitions and standard-form equations of circles, parabolas, ellipses and hyperbolas, and you use completing the square to convert expanded equations into standard form. You find centers, vertices, foci, directrices and asymptotes, and you graph each curve. The unit ends with solving nonlinear systems, such as a line and a circle or two conics, using substitution and elimination.
A circle keeps a fixed distance from its center. A parabola stays equally far from a focus and a directrix line. An ellipse keeps the sum of distances to two foci constant, and a hyperbola keeps the absolute difference of those distances constant.
In you can see the center and radius right away. The same goes for the vertex and in a parabola, or , and the center in an ellipse or hyperbola. When an equation is expanded, completing the square brings it back to standard form.
Substitution and elimination still work, but the result is usually a quadratic, so you can get 0, 1, 2 or more solutions. A line and a nondegenerate conic meet at most 2 times, and two distinct nondegenerate conics meet at most 4 times.
A general second-degree equation has the form . It can describe a conic, a degenerate graph such as a point or a pair of lines, or no real points. When and the graph is a nondegenerate conic, gives a circle, nonzero and with the same sign but different values give an ellipse, opposite signs give a hyperbola, and exactly one nonzero squared coefficient gives a parabola. An term can indicate a rotated conic.
A circle is every point at distance r from a center (h,k), which gives (x-h)^2+(y-k)^2=r^2.
An ellipse is every point where the sum of the distances to two foci is constant.
A solution to a system is a point that lies on both graphs.
A circle is every point at distance from a center , which gives . That equation is the distance formula squared. A parabola is every point that is the same distance from a fixed point, the focus, and a fixed line, the directrix.
For a parabola with vertex that opens up or down, use . The focus is units from the vertex along the axis of symmetry, and the directrix is units away on the opposite side. It opens up when and down when . If is the squared variable, , the parabola opens right when and left when . This matches vertex form with .
Circles often appear expanded, as in . Complete the square in and in to get standard form back.
Worked example. Find the center and radius of . Group terms and move the constant, which gives . Half of is and . Half of is and . Add 9 and 4 to both sides to get . The center is and the radius is .
Check yourself: Where are the focus and directrix of ? (Answer: , so . The vertex is , the focus is , and the directrix is .)
An ellipse is every point where the sum of the distances to two foci is constant. A hyperbola is every point where the absolute difference of those distances is a positive constant. The standard-form equation for an ellipse uses a plus sign between its squared terms, while a hyperbola uses a minus sign.
An ellipse centered at is or the same with the denominators switched. The larger denominator is always , and it sits under the variable of the major axis. The foci satisfy , so they lie inside the curve.
A hyperbola has a minus sign between the fractions. The positive term tells you the direction the branches open, and is always under that term, even if it is the smaller number. The foci satisfy , so they lie beyond the vertices. The asymptotes pass through the center along the diagonals of a rectangle centered there, with side lengths along the transverse axis and along the other axis.
Worked example. For , and , so and . The vertices are and the foci are . For , and , so and . The vertices are , the foci are , and the asymptotes are .
Check yourself: Which way does open, and where are its foci? (Answer: The term is positive, so it opens up and down. and , so and the foci are .)
| Feature | Ellipse | Hyperbola |
|---|---|---|
| Distance definition | Sum of distances to foci is constant | Absolute difference of distances to foci is constant |
| Sign between terms | Plus | Minus |
| Where $a^2$ goes | Under the larger denominator | Under the positive term |
| Focus formula | $c^2=a^2-b^2$ | $c^2=a^2+b^2$ |
| Foci location | Inside the curve, $c<a$ | Beyond the vertices, $c>a$ |
| Eccentricity | $0<e<1$ | $e>1$ |
| Asymptotes | None | Two lines through the center |
A solution to a system is a point that lies on both graphs. Sketching the curves first tells you roughly how many solutions to expect and where they are.
For a line and a conic, solve the linear equation for one variable and substitute it into the conic. The result can be quadratic or linear, and a nondegenerate conic and a line have 0, 1 or 2 intersection points. If the result is quadratic, a discriminant of 0 gives one repeated solution and a tangent line. A linear result can also give one intersection without tangency.
For two conics with only squared terms, elimination is usually faster. For example, adding and gives , so and then , for four solutions. Two distinct nondegenerate conics can meet at up to 4 points.
Worked example. Solve and . Substitute to get . Expand to get , then . Factor as , so or . Plug back into to get and . Both check, since and .
In word problems, throw out solutions that don't fit the context, like a negative distance.
Check yourself: How many times does the line meet the circle ? (Answer: Once. Substituting gives , so the only point is and the line is tangent to the circle.)
| Term | Definition |
|---|---|
| Conic section | A curve formed where a plane intersects a double cone, namely a circle, parabola, ellipse or hyperbola. |
| Standard form | A version of a conic's equation, such as $(x-h)^2+(y-k)^2=r^2$, that shows its center or vertex and its dimensions directly. |
| Focus | A fixed point used to define a conic, with one for a parabola and two (called foci) for an ellipse or hyperbola. |
| Directrix | The fixed line that, together with the focus, defines a parabola as all points equally distant from both. |
| Vertex | A point where a conic crosses its main axis of symmetry, such as the turning point of a parabola or an endpoint of an ellipse's major axis. |
| Co-vertex | An endpoint of an ellipse's minor axis, located $b$ units from the center. |
| Asymptote | A line through the center of a hyperbola that the branches get closer and closer to without reaching. |
| Eccentricity | A number that describes a conic's shape. For a circle, ellipse or hyperbola, $e=\frac{c}{a}$; for a parabola, $e=1$. |
| General form of a conic | The expanded second-degree equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$. It can describe a conic, a degenerate graph or no real points; an $xy$ term can indicate a rotated conic. |
The ellipse formula sticks in your head and gets reused. Tie the sign to the equation. A plus sign between the fractions means the ellipse formula , and a minus sign means the hyperbola formula .
Textbooks often show under , so it is easy to treat that as a rule. For an ellipse, is the larger denominator wherever it sits. For a hyperbola, is under the positive term, even when it is smaller than .
In , the center is , not . Standard form uses , so rewrite as to read correctly. Also remember that the right side is , so the radius here is 3, not 9.
When you take a square root, like , both and count. Finding is also only half the job, because each can pair with one or two values. Substitute every back in and check each point in both original equations.