Conic Sections Exercises and Problems
Three seismic stations across the country record an earthquake at the exact same moment. We know their positions and nothing else. Where is the epicenter? To answer, you have to recognize which curve is formed by the points that are the same distance from all three stations. That is what you practice here: you already know the conic sections one by one, and now it's time to tell them apart, set them up from a word problem, and solve complete problems.
Which conic is it? Look at the equation
If the equation is expanded, the coefficients of $x^2$ and $y^2$ tell you almost everything.
When the equation is already in reduced form, the elements can be read directly:
| Conic | Equation (center at the origin) | Elements |
|---|---|---|
| Ellipse | $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, with $a>b$ | $c^2=a^2-b^2$, $e=\frac{c}{a}<1$, foci $(\pm c, 0)$ |
| Hyperbola | $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ | $c^2=a^2+b^2$, $e=\frac{c}{a}>1$, asymptotes $y=\pm\frac{b}{a}x$ |
| Parabola | $y^2=4px$ | focus $(p, 0)$, directrix $x=-p$, vertex $(0, 0)$ |
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 $y^2 = -20x$.
Find the lengths 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 lengths 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.
General equation: completing the square
When the conic is translated, $Dx$ and $Ey$ terms appear. Group the $x$ terms on one side and the $y$ terms on the other, and complete the square to reach the standard form.
Determine the coordinates of the center and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
Determine the coordinates of the center and the radius of the circle with general equation $x^2 + y^2 - 8x + 10y - 8 = 0$.
A circle has its center at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
A circle has its center at point $C(-3, 5)$ and passes through the origin. Write its standard equation and its general equation.
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.
Loci: look at the word problem
If there is no equation, look at the condition that the point $P(x, y)$ satisfies:
| The condition says… | The locus is… |
|---|---|
| constant distance from a point | circle |
| sum of distances to two points is constant | ellipse |
| difference (in absolute value) of distances to two points is constant | hyperbola |
| equal distance from a point and a line | parabola |
| distance to a point $=e\times$ distance to a line | ellipse if $e<1$, parabola if $e=1$, hyperbola if $e>1$ |
| equal distance from two points | perpendicular bisector (a line) |
| equal distance from two lines | angle bisectors (lines) |
| distance to $A$ $=k\times$ distance to $B$, with $k\neq1$ | circle |
The method is always the same: write the condition using $P(x, y)$, use the distance formula, square both sides, and simplify.
With the angle bisector, we use the distance from a point to a line, $d=\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$. The distances to the two lines are set equal and the absolute values are removed with $\pm$, which gives the two bisectors:
$$\frac{|A_1x+B_1y+C_1|}{\sqrt{A_1^2+B_1^2}}=\frac{|A_2x+B_2y+C_2|}{\sqrt{A_2^2+B_2^2}}$$Focus, directrix, and eccentricity
The parabola has a focus and a directrix. The ellipse and the hyperbola also have directrices: each focus $(\pm c, 0)$ has an associated vertical directrix.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ and $B(-4, 5)$.
Find the equation of the perpendicular bisector of the line segment with endpoints $A(2, -3)$ 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 $s: 5x + 12y - 10 = 0$.
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its…
Identify the locus of points in the plane whose difference of distances, in absolute value, to the points $F_1(5, 0)$ and $F_2(-5, 0)$ is always equal to $8$. Write its standard equation.
An underwater drone moves in such a way that its distance to a fixed buoy at $A(-2, 0)$ is always double 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 double its distance to another fixed buoy at $B(4, 0)$. Find the equation of the drone's path, simplify it, and identify the type of geometric curve it describes.
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 sketch it.
Find the equation of the locus of points in the plane whose distance from the fixed point $F(4, 0)$ is equal to half their distance from the vertical line $x = 16$.…
Find the equation of the locus of points in the plane whose distance from the fixed point $F(4, 0)$ is equal to half their distance from the vertical line $x = 16$. Identify the resulting conic, find its semi-axes, and sketch it, illustrating the directrix property.
Real-world problems with several steps
Real problems almost never give you $a$, $b$, $c$, or $p$. First you have to translate the problem into those data.
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Place the axes conveniently (center or vertex at the origin).
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Write down which points you know, with their coordinates.
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Choose the conic and its equation.
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Substitute the data to find the parameters.
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Answer the question with the units from the problem.
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometers). Assuming that the seismic waves travel at the same speed…
Three seismic stations detect an earthquake at coordinates $A(1, 4)$, $B(-3, 2)$ and $C(5, -2)$ (in kilometers). Assuming that the seismic waves travel at the same speed and arrive at the same time at all three stations, find the coordinates of the epicenter and the radius of the affected area (distance to the stations).
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 lies on 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 at a height of $50 \text{ m}$…
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 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 its center coincides with the center of the narrowest part.
The main cable of a suspension bridge forms a parabola. The two towers supporting it are $120 \text{ m}$ apart and have a height of $40 \text{ m}$ above the bridge deck.…
The main cable of a suspension bridge forms a parabola. 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 $30 \text{ m}$ from the center of the bridge.
Conics and lines: intersections and tangents
Intersection points
Solve for one variable in the line and substitute into the equation of the conic. You get a quadratic equation whose discriminant $\Delta$ decides: $\Delta>0$, two points (secant); $\Delta=0$, one point (tangent); $\Delta<0$, none (exterior).
Tangents
There are two situations and two tools:
Tangents from an exterior point
If the point is not on the curve, the tangent is written with an unknown slope $m$: $y-y_0=m(x-x_0)$. For a circle, we require the distance from the center to the line to equal the radius.
This same idea works for a parameter in a line: a ship is "detected" if the line intersects the circle, that is, if the distance from the center to the line is less than or equal to the radius.
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of…
Find the points of intersection between the circle $x^2 + y^2 - 4x - 2y - 20 = 0$ and the line $x - y - 2 = 0$. Graph the situation, showing both loci and the points of intersection.
You can do it on paper and send a photo with your answer.
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path given by the equation…
A marine radar has a detection range modelled by the circle $x^2 + y^2 - 4x + 6y - 12 \leq 0$. A ship follows a straight-line path 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 along its path.
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 x-coordinate $x = 3$ and…
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 x-coordinate $x = 3$ and a positive y-coordinate. Find the equation of the tangent line to the mirror at that point.
From a control station located at 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 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 straight-line paths 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. A ray of light leaves the…
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. 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 that the incident ray makes with this tangent.
Intersection between curves
Two circles: radical axis
If you subtract the general equations of two circles, $x^2$ and $y^2$ disappear and you are left with a line, the radical axis. It passes through the intersection points, if there are any.
They are secant when $|r1-r2| Solve the system by substitution. Watch out: some solutions of the resulting equation may not be valid. A rectangular hyperbola ($a=b$) whose asymptotes are the coordinate axes has equation $xy=k$. To find $k$, just substitute a point. If $k>0$, the branches are in quadrants I and III, and the focal axis is the line $y=x$: vertices $(\sqrt k, \sqrt k)$ and $(-\sqrt k, -\sqrt k)$; $a=b=\sqrt{2k}$ and $c=a\sqrt2=2\sqrt k$; foci $(\sqrt{2k}, \sqrt{2k})$ and $(-\sqrt{2k}, -\sqrt{2k})$. Check: the focus is at distance $\sqrt{2k+2k}=2\sqrt k=c$ from the origin. A comet follows a parabolic path 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 paths. Two telecommunications networks have coverage areas modelled 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 intersect at two points, and calculate the exact distance between their points of intersection. A subatomic particle moves in a magnetic field, describing an equilateral hyperbolic path whose asymptotes coincide with the coordinate axes. Knowing that the particle passes through the point $P(4, 2)$, solve: You can do it on paper and send a photo with your answer.A parabola and a hyperbola
Rectangular hyperbola with asymptotes along the axes
A comet follows a parabolic path given by the equation $y^2 = 8x$. An asteroid follows a rectangular hyperbolic orbit described by $x^2 - y^2 = 9$. Find the exact…
Two telecommunications networks have coverage areas modelled 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…
A subatomic particle moves in a magnetic field, describing an equilateral hyperbolic path whose asymptotes coincide with the coordinate axes. Knowing that the particle…