Conics Problem Solving
The Golden Gate Bridge's main cables sag in a parabola, a satellite loops around Earth in an ellipse, and a cooling tower flares out in a hyperbola. On a test, nobody tells you which conic you are facing. Your first job is to figure that out, from an equation or from a word description. Your second job is to solve the whole problem. This page gives you a toolkit for both.
Reading the Equation
Start with the general equation $Ax^2 + By^2 + Dx + Ey + F = 0$. The coefficients $A$ and $B$ of the squared terms tell you the conic.
To get the center, radius, vertices, and so on, complete the square in $x$ and in $y$.
Shortcut for a circle $x^2 + y^2 + Dx + Ey + F = 0$: the center is $\left(-\tfrac{D}{2}, -\tfrac{E}{2}\right)$ and $r^2 = \tfrac{D^2}{4} + \tfrac{E^2}{4} - F$.
Determine the coordinates of the center and the length of the radius of the circle whose general equation is $x^2 + y^2 - 8x + 10y - 8 = 0$.
Determine the coordinates of the center and the length of the radius of the circle whose general equation is $x^2 + y^2 - 8x + 10y - 8 = 0$.
Given the conic with general equation $9x^2 + 25y^2 - 54x + 200y + 256 = 0$. Classify it by completing the square and find the coordinates of its center, its vertices,…
Given the conic with general equation $9x^2 + 25y^2 - 54x + 200y + 256 = 0$. Classify it by completing the square and find the coordinates of its center, its vertices, its foci and the value of its eccentricity.
Standard Forms and Their Parts
Once an equation is in standard form, every part of the conic can be read off. Here is the reference for conics centered (or with vertex) at the origin. To move the center to $(h, k)$, replace $x$ with $x - h$ and $y$ with $y - k$.
| Conic | Standard form | Key facts |
|---|---|---|
| Circle | $x^2 + y^2 = r^2$ | center $(0, 0)$, radius $r$ |
| Ellipse, horizontal | $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, with $a > b > 0$ | vertices $(\pm a, 0)$; foci $(\pm c, 0)$ with $c^2 = a^2 - b^2$; $e = \dfrac{c}{a} < 1$ |
| Ellipse, vertical | $\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1$, with $a > b > 0$ | vertices $(0, \pm a)$; foci $(0, \pm c)$ with $c^2 = a^2 - b^2$ |
| Hyperbola, horizontal | $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ | vertices $(\pm a, 0)$; foci $(\pm c, 0)$ with $c^2 = a^2 + b^2$; asymptotes $y = \pm\dfrac{b}{a}x$; $e = \dfrac{c}{a} > 1$ |
| Hyperbola, vertical | $\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1$ | vertices $(0, \pm a)$; foci $(0, \pm c)$; asymptotes $y = \pm\dfrac{a}{b}x$ |
| Parabola, opens left/right | $y^2 = 4px$ | vertex $(0, 0)$; focus $(p, 0)$; directrix $x = -p$ |
| Parabola, opens up/down | $x^2 = 4py$ | vertex $(0, 0)$; focus $(0, p)$; directrix $y = -p$ |
For a parabola, $p > 0$ opens toward the positive direction and $p < 0$ opens the other way.
Why is the ellipse $c^2 = a^2 - b^2$ but the hyperbola $c^2 = a^2 + b^2$? In the ellipse, $a$ is the hypotenuse of the triangle above, so it is the biggest. In the hyperbola, the foci lie farther from the center than the vertices, so $c$ is the biggest.
Find the length of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
Find the length of the semi-axes, the coordinates of the foci, and the eccentricity of the ellipse with equation $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine its vertices, its foci, and the equations of its asymptotes.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
Find the coordinates of the vertex, the focus, and the equation of the directrix of the parabola $y^2 = -20x$.
Reading a Locus Description
A locus is the set of all points that satisfy a condition. When a problem has no equation, look at the condition on the distances.
The last two rows of the chart need a little algebra: write $P(x, y)$, translate the condition into an equation, square, and simplify.
Identify the locus of the points in the plane whose difference in distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write…
Identify the locus of the points in the plane whose difference in distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its reduced equation.
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always twice its distance to another fixed buoy at $B(4, 0)$. Find the…
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always twice its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's trajectory, simplify it, and identify what type of geometric curve it describes.
Slope Products and the Focus-Directrix Rule
Two more locus conditions appear often.
Product of slopes
Take $A(-a, 0)$ and $B(a, 0)$. For $P(x, y)$, the slope of $PA$ is $\dfrac{y}{x + a}$ and the slope of $PB$ is $\dfrac{y}{x - a}$. If the product of these slopes is a constant $m$, then $\dfrac{y^2}{x^2 - a^2} = m$.
Focus-directrix rule
A conic is the set of points $P$ with $d(P, F) = e \cdot d(P, \ell)$, where $F$ is a focus, $\ell$ is a line called the directrix, and $e$ is the eccentricity.
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is…
Find the locus of the points in the plane $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(-4, 0)$ and $B(4, 0)$ is equal to $-\frac{9}{16}$. Identify the resulting conic, calculate its eccentricity and represent it graphically.
Find the equation of the locus of the points in the plane whose distance to the fixed point $F(4, 0)$ is equal to half their distance to the vertical line $x = 16$.…
Find the equation of the locus of the points in the plane whose distance to the fixed point $F(4, 0)$ is equal to half their distance to the vertical line $x = 16$. Identify the resulting conic, find its semi-axes and represent it graphically illustrating the directrix property.
Perpendicular Bisectors and Angle Bisectors
Two loci are lines, not conics, and they show up everywhere in conic problems.
When the two lines have different denominators, such as $5$ and $13$, keep both: multiply through so that $13(\text{first}) = \pm 5(\text{second})$.
Find the equation of the perpendicular bisector of the line segment whose endpoints are the points $A(2, -3)$ and $B(-4, 5)$.
Find the equation of the perpendicular bisector of the line segment whose endpoints are the points $A(2, -3)$ and $B(-4, 5)$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Write the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 12 = 0$ and $s: 5x + 12y - 10 = 0$.
Circles from Geometric Conditions
A circle is fixed by its center and radius. Many problems give you those indirectly. The equation $(x - h)^2 + (y - k)^2 = r^2$ is the standard form of a circle, and the expanded version $x^2 + y^2 + Dx + Ey + F = 0$ is the general form.
Circle through three points. The center must be equidistant from all three points, so it lies on the perpendicular bisectors of the chords. It is the intersection of two of them.
There is a second method. Plug each point into $x^2 + y^2 + Dx + Ey + F = 0$ and solve the three linear equations for $D$, $E$, $F$. For the points above, you get $D = -6$, $E = -8$, $F = 5$, which is the same circle expanded.
A circle has its center at the point $C(-3, 5)$ and passes through the origin of coordinates. Write its reduced equation and its general equation.
A circle has its center at the point $C(-3, 5)$ and passes through the origin of coordinates. Write its reduced equation and its general equation.
Three seismic stations detect an earthquake at the coordinates $A(1, 4)$, $B(-3, 2)$, and $C(5, -2)$ (in miles). Assuming that the seismic waves travel at the same speed…
Three seismic stations detect an earthquake at the coordinates $A(1, 4)$, $B(-3, 2)$, and $C(5, -2)$ (in miles). Assuming that the seismic waves travel at the same speed and arrive at the three stations at the same time, find the coordinates of the epicenter and the radius of the affected area (distance to the stations).
Modeling Word Problems
Real problems never hand you $a$, $b$, $c$, or $p$. You build them from the story.
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Sketch the situation.
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Place the axes so the equation is as simple as possible: origin at the center of an ellipse or hyperbola, or at the vertex of a parabola.
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Translate each given number into $a$, $b$, $c$, or $p$, or into a point on the curve.
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Solve for the unknowns and write the equation.
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Answer in context, with units.
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is…
A satellite describes an elliptical orbit around a planet, which is located at one of its foci. The minimum distance from the satellite to the planet (periapsis) is $3 \text{ Mm}$ and the maximum distance (apoapsis) is $9 \text{ Mm}$. Determine the equation of the orbit and its eccentricity, assuming that the center of the ellipse is at the origin and the focal axis coincides with the x-axis.
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is located at a height of…
The cross-section of a nuclear cooling tower has the shape of a hyperbola. The narrowest part of the tower is $40 \text{ m}$ wide and is located at a height of $50 \text{ m}$ above the ground. If the base of the tower has a width of $80 \text{ m}$, find the equation of the hyperbola assuming that its center coincides with the center of the narrowest part.
The main cable of a suspension bridge adopts a parabolic shape. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the…
The main cable of a suspension bridge adopts a parabolic shape. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck. The lowest point of the cable is $10 \text{ m}$ above the deck. Determine the equation of the parabola and calculate the height of the cable at $30 \text{ m}$ from the center of the bridge.
Lines Meeting Conics
To find where a line meets a conic, substitute the line into the conic's equation. You get a quadratic whose discriminant counts the intersection points.
For a circle, there is a faster test for "does the line touch it?" Compare the distance from the center to the line with the radius: $d < r$ means two points, $d = r$ means tangent, and $d > r$ means none. A "detected somewhere along the path" question about a disk (a circle with its inside) means $d \le r$.
If the circle is not centered at the origin, complete the square first to find its center and radius.
A maritime radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a rectilinear trajectory given by the equation…
A maritime radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a rectilinear trajectory given by the equation $3x + 4y + k = 0$. Determine the values of the parameter $k$ for which the ship is detected by the radar at some point in its trajectory.
Calculate the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Represent the situation graphically showing both…
Calculate the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Represent the situation graphically showing both loci and the points of intersection.
A comet follows a parabolic trajectory given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact…
A comet follows a parabolic trajectory given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact coordinates of the points of intersection of both trajectories.
Tangent Lines
A tangent line touches a conic at exactly one point. There are two situations.
Tangent at a point on the conic
Tangents from an outside point
A point outside the conic has two tangents. For a circle: write a line through the point $P$ with unknown slope $m$, and require the distance from the center to the line to equal the radius. Solve for $m$. Do not forget the vertical line through $P$, which has no slope.
Angle between two lines
If two lines have slopes $m_1$ and $m_2$, the angle $\theta$ between them satisfies $\tan\theta = \left|\dfrac{m_2 - m_1}{1 + m_1 m_2}\right|$. For example, slopes $1$ and $3$ give $\tan\theta = \tfrac{2}{4} = 0.5$, so $\theta \approx 26.6^\circ$. If one line is vertical, use its inclination of $90^\circ$ and the other line's inclination $\arctan m$: the angle is the difference.
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has an x-coordinate of…
A laser beam strikes an elliptical mirror whose shape is given by the equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The point of incidence has an x-coordinate of $x = 3$ and a positive y-coordinate. Find the equation of the tangent line to the mirror at this point.
From a control station located at the point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation…
From a control station located at the point $P(0, 7)$, laser signals are sent to a satellite whose circular orbit at that instant is given by the equation $x^2 + y^2 - 4x + 2y - 15 = 0$. Find the equations of the two rectilinear trajectories tangent to the orbit from the control station.
A car headlight has a parabolic cross-section. The bulb is located at the focus $F(3, 0)$ and the vertex of the parabola is at the origin of coordinates. A ray of light…
A car headlight has a parabolic cross-section. The bulb is located at the focus $F(3, 0)$ and the vertex of the parabola is at the origin of coordinates. A ray of light leaves the bulb, strikes the parabola at a point with y-coordinate $y = 6$ and is reflected. Calculate the equation of the tangent line at the point of incidence and find the angle formed by the incident ray and this tangent.
Radical Axis and Rectangular Hyperbolas
Two last tools complete the toolkit.
Radical axis of two circles
Rectangular hyperbola $xy = k$
When the asymptotes of a hyperbola are the coordinate axes, the hyperbola is rectangular ($a = b$) and its equation is $xy = k$. It is the ordinary hyperbola rotated $45^\circ$. For $k > 0$ the branches are in quadrants I and III, and the transverse axis is the line $y = x$.
Two telecommunications networks have coverage areas modeled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of…
Two telecommunications networks have coverage areas modeled by the circles $C_1: x^2 + y^2 - 6x + 8y = 0$ and $C_2: x^2 + y^2 - 2x - 4y + 1 = 0$. Find the equation of the radical axis, prove analytically that the circles are secant and calculate the exact distance between their points of intersection.
A subatomic particle moves in a magnetic field describing a rectangular hyperbolic trajectory whose asymptotes coincide with the coordinate axes. Knowing that the…
A subatomic particle moves in a magnetic field describing a rectangular hyperbolic trajectory whose asymptotes coincide with the coordinate axes. Knowing that the particle passes through the point $P(4, 2)$, find the equation of the trajectory. Determine the exact coordinates of the vertices and the foci, and represent the curve graphically.
Summary
Back to the start: a satellite orbit, a bridge cable, and a cooling tower are an ellipse, a parabola, and a hyperbola. You recognize them by what the problem gives you (a sum of distances, a distance to a line, a difference), and you finish by turning the story into $a$, $b$, $c$, or $p$.