Choose the type of curve and enter the denominators (ellipse or hyperbola) or the coefficient of the linear term (parabola). The calculator finds the foci, vertices and eccentricity, plus the asymptotes of a hyperbola or the directrix of a parabola.
Enter numbers only. Decimals and fractions such as 3/4 are OK. For the denominators a² and b², enter the number written under x² or y², not the value of a or b (for x²/9, enter 9). A blank counts as 1 (a denominator or coefficient that is not written is 1).
Result and graph
Choose the type of curve in the fields on the left, enter the numbers and press "Calculate". The foci, eccentricity and other results will appear here with a graph.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the denominators or the coefficient, and you get the foci, vertices and eccentricity of the ellipse \(\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1\), the hyperbola \(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = \pm 1\) or the parabola \(y^{2} = 4px\) (and \(x^{2} = 4py\)) on the spot
- Answers are shown both as exact values with \(\sqrt{\ }\), such as \(\left(\pm\sqrt{5},\ 0\right)\), and as decimals. You can also see the steps that find the foci from \(c^{2} = a^{2} - b^{2}\)
- For a hyperbola you also get the asymptotes, and for a parabola the directrix. The curve, foci, asymptotes, directrix and vertices are drawn on a graph
- You can check the textbook definitions on the graph: "the sum (or difference) of the distances to the two foci is constant" and "the same distance from the focus and the directrix"
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page
This page works with standard forms centered at the origin (for a parabola, with the vertex at the origin). Shifted equations such as \(\dfrac{(x-1)^{2}}{9} + \dfrac{y^{2}}{4} = 1\) are not supported. Ellipses that are wide or tall, hyperbolas that open left-right or up-down, and parabolas that open right, left, up or down are all supported.
What is this calculation used for?
Predicting the orbits of planets and comets (astronomy and space flight)
Planets travel in elliptical orbits with the Sun at one focus (Kepler's first law). Earth's orbit has an eccentricity of about 0.017, very close to a circle, while Halley's Comet follows a long, thin ellipse with an eccentricity of about 0.97 and comes back to the Sun about every 76 years.
The same math is the basis for designing the paths of satellites and space probes, and the shape of an orbit (ellipse, parabola or hyperbola) is classified by its eccentricity. When a probe escapes a planet's gravity and flies away, its path is a hyperbola.
Designing satellite dishes and headlights (radio and optics)
A surface made by spinning a parabola around its axis (a paraboloid) has this property: radio waves or light coming in parallel to the axis all reflect to the focus. A satellite TV dish gathers weak signals from far above into the receiver placed at the focus.
Car headlights and searchlights use it the other way around: light from a source at the focus reflects into a straight beam parallel to the axis. The position of the focus (the \(p\) this page calculates) is the key to the design.
Breaking up kidney stones with shock waves (medicine)
An ellipse has this property: a wave that starts at one focus reflects off the ellipse and gathers at the other focus. Extracorporeal shock wave lithotripsy (ESWL) creates a shock wave at one focus of an elliptical reflector and lines up the other focus with the kidney stone inside the body, breaking the stone without surgery.
Finding the exact position of the foci directly affects how precise the treatment is.
Secondary mirrors in telescopes and antennas (optical design)
A Cassegrain reflecting telescope combines a paraboloid primary mirror with a hyperboloid secondary mirror. It uses a property of the hyperboloid, "light heading toward one focus reflects and gathers at the other focus", to guide the light to the eyepiece at the back of the tube.
Large radio telescopes and satellite communication antennas mostly use the same design, so the focus properties of ellipses, hyperbolas and parabolas work together inside one device.
Finding a position from differences in signal arrival time (navigation)
Some positioning systems use the very definition of a hyperbola: "the points whose distances to two fixed points differ by a constant". In hyperbolic navigation such as LORAN, a ship or aircraft measured the difference in arrival time of signals from two stations to find which hyperbola it was on, then found its position where that hyperbola crossed another one from a different pair of stations.
GPS has replaced it, but the idea of narrowing down a position from time differences lives on, for example in estimating the location of a cell phone during an emergency call.
Curved structures in building and civil engineering (cooling towers and cables)
Many of the huge cooling towers at power plants have the shape of a hyperboloid, made by spinning a hyperbola. This curved surface can be built just from straight steel members set at an angle, yet it is strong and saves material.
The main cable of a suspension bridge is a parabola when the weight of the deck is spread evenly along the horizontal. (A cable hanging only under its own weight forms a different curve that looks similar, called a catenary.) In design drawings, standard-form equations like the ones on this page define these shapes.
Formulas and figures
Foci of an ellipse (\(\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1\), \(a > b > 0\))
Figure
Standard notation
(the usual math form)
\(c^{2}\)
\(=\)
\(a^{2}\)
\(-\)
\(b^{2}\)
In words
(symbols replaced with words)
③
\(c^{2}\): square of the distance from the center to a focus
\(=\)
①
\(a^{2}\): \(x^{2}\) denominator (half the major axis, squared)
\(-\)
②
\(b^{2}\): \(y^{2}\) denominator (half the minor axis, squared)
The formula in words
①
From the
\(x^{2}\) denominator \(a^{2}\)
②
subtract the
\(y^{2}\) denominator \(b^{2}\)
③
and you get the
square of the distance \(c\) from the center to a focus
(the foci are the two points \((\pm c,\ 0)\) on the \(x\)-axis)
Quick example
The foci of \(\dfrac{x^{2}}{25} + \dfrac{y^{2}}{9} = 1\) (\(a^{2} = 25,\ b^{2} = 9\)) are
distance to a focus \(c\), squared
\(=\)
\(x^{2}\) denominator (25)
\(-\)
\(y^{2}\) denominator (9)
\(c^{2} = 25 - 9 = 16\)
\(c = \sqrt{16} = 4\)
\(F(4,\ 0),\quad F'(-4,\ 0)\)
Key idea
An ellipse is the set of points whose distances to the two foci always add up to the same number (\(= 2a\)). Put two pins in a board, loop a string around them, keep the string tight with a pencil and move it around: you draw an ellipse. This is exactly the definition.
If the \(x^{2}\) denominator is larger, the ellipse is wide, and the foci are at \((\pm c,\ 0)\) on the \(x\)-axis. If the \(y^{2}\) denominator is larger, the ellipse is tall, \(c^{2} = b^{2} - a^{2}\), and the foci move to \((0,\ \pm c)\) on the \(y\)-axis. Remember "larger denominator minus smaller denominator", and you will not get the direction wrong.
If \(a^{2} = b^{2}\), then \(c = 0\), and the two foci meet at the center: the ellipse is a circle. A circle is a special case of an ellipse.
Foci and asymptotes of a hyperbola (\(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1\))
Figure
Standard notation
(the usual math form)
\(c^{2}\)
\(=\)
\(a^{2}\)
\(+\)
\(b^{2}\)
In words
(symbols replaced with words)
③
\(c^{2}\): square of the distance from the center to a focus
\(=\)
①
\(a^{2}\): \(x^{2}\) denominator (distance to a vertex, squared)
\(+\)
②
\(b^{2}\): \(y^{2}\) denominator
The formula in words
①
To the
\(x^{2}\) denominator \(a^{2}\)
②
add the
\(y^{2}\) denominator \(b^{2}\)
③
and you get the
square of the distance \(c\) from the center to a focus
(the foci are the two points \((\pm c,\ 0)\) on the \(x\)-axis)
Quick example
The foci of \(\dfrac{x^{2}}{9} - \dfrac{y^{2}}{16} = 1\) (\(a^{2} = 9,\ b^{2} = 16\)) are
distance to a focus \(c\), squared
\(=\)
\(x^{2}\) denominator (9)
\(+\)
\(y^{2}\) denominator (16)
\(c^{2} = 9 + 16 = 25\)
\(c = \sqrt{25} = 5\)
\(F(5,\ 0),\quad F'(-5,\ 0)\)
Key idea
A hyperbola is the set of points whose distances to the two foci always differ by the same number (\(= 2a\)). An ellipse uses \(c^{2} = a^{2} - b^{2}\) (subtraction), and a hyperbola uses \(c^{2} = a^{2} + b^{2}\) (addition): just one sign is different. The reason is that in an ellipse the foci are inside the vertices (\(c < a\)), while in a hyperbola the foci are outside the vertices (\(c > a\)), so \(c^{2}\) is larger than \(a^{2}\).
A hyperbola has two lines that the curve gets closer and closer to, called asymptotes: \(y = \pm\dfrac{b}{a}x\) (for the example hyperbola, \(y = \pm\dfrac{4}{3}x\)). When graphing, the usual method is to draw the asymptotes first and then sketch the curve along them.
If the right side is \(-1\) (\(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = -1\)), the hyperbola opens up and down. The foci are at \((0,\ \pm c)\) on the \(y\)-axis, the vertices are at \((0,\ \pm b)\), and the difference of the distances becomes \(2b\). The way to find \(c\) (addition) and the asymptotes stay the same.
Focus and directrix of a parabola (\(y^{2} = 4px\))
Figure
Standard notation
(the usual math form)
\(PF\)
\(=\)
\(PH\)
In words
(symbols replaced with words)
②
\(PF\): distance to the focus \(F\)
\(=\)
①
\(PH\): distance to the directrix
The formula in words
①
A parabola is the set of points \(P\) where the
distance to the directrix \(PH\)
②
and the
distance to the focus \(F\), \(PF\),
are always equal
Quick example
Comparing \(y^{2} = 8x\) with the standard form \(y^{2} = 4px\) gives \(4p = 8\), so
distance to the focus \((2,\ 0)\)
\(=\)
distance to the directrix \(x = -2\)
\(4p = 8\)
\(p = \dfrac{8}{4} = 2\)
Key idea
The focus of \(y^{2} = 4px\) is \((p,\ 0)\) on the \(x\)-axis, and the directrix is the vertical line \(x = -p\). If \(p > 0\), the parabola opens to the right; if \(p < 0\), it opens to the left.
Swapping \(x\) and \(y\) gives \(x^{2} = 4py\), a parabola that opens up or down, with focus \((0,\ p)\) and the horizontal directrix \(y = -p\). The familiar graph of \(y = ax^{2}\) is in the same family: rewrite it as \(x^{2} = \dfrac{1}{a}y\), and it has the focus \(\left(0,\ \dfrac{1}{4a}\right)\) and a directrix.
The coefficient is written as \(4p\) because then the focus and directrix come out in the clean forms \((p,\ 0)\) and \(x = -p\). When you are given something like \(y^{2} = 8x\), the first step is to find \(p\) from \(4p = 8\).
Eccentricity \(e\) (a number that describes the shape of the curve)
Standard notation
(the usual math form)
\(e\)
\(=\)
\(c\)
\(\div\)
\(a\)
In words
(symbols replaced with words)
③
\(e\): eccentricity
\(=\)
①
\(c\): distance from the center to a focus
\(\div\)
②
\(a\): distance from the center to a vertex
The formula in words
①
Take the
distance \(c\) from the center to a focus
②
divide it by the
distance \(a\) from the center to a vertex (the end of the longer axis)
③
and you get the
eccentricity \(e\)
Quick example
The eccentricity of \(\dfrac{x^{2}}{25} + \dfrac{y^{2}}{9} = 1\) (\(a = 5,\ c = 4\)) is
eccentricity \(e\)
\(=\)
distance to a focus (4)
\(\div\)
distance to a vertex (5)
\(e = \dfrac{4}{5} = 0.8\)
Key idea
Eccentricity is a number that tells how far a curve is from being a circle. A circle has \(e = 0\), an ellipse has \(0 < e < 1\) (closer to 0 is rounder, closer to 1 is longer and thinner), a parabola has exactly \(e = 1\), and a hyperbola has \(e > 1\).
Ellipses, hyperbolas and parabolas look like unrelated curves at first, but this one number arranges them in a smooth sequence: circle → ellipse → parabola → hyperbola. This is one reason the three are grouped together as conic sections.
For a tall ellipse or a hyperbola that opens up and down, the longer axis is the \(y\)-axis, so you divide by \(b\) (the distance from the center to a vertex) instead. Remember "distance to a focus ÷ distance to a vertex", and you will get it right in any direction.
The distance \(c\) from the center to a focus is found with \(c^{2} = a^{2} - b^{2}\) for an ellipse and \(c^{2} = a^{2} + b^{2}\) for a hyperbola. The parabola \(y^{2} = 4px\) has the focus \((p,\ 0)\) and the directrix \(x = -p\). The three curves are a family (the conic sections) defined by distances to a focus, and their shapes are sorted by the eccentricity \(e\): ellipse \(e < 1\), parabola \(e = 1\), hyperbola \(e > 1\).
Symbols and terms
Symbols
| \(a,\ b\) | a, b | Positive numbers in the denominators of the standard forms of the ellipse and hyperbola (as \(a^{2}\) and \(b^{2}\)). For an ellipse, \(a\) is the radius in the \(x\) direction and \(b\) the radius in the \(y\) direction (each is half the major or minor axis). For a hyperbola, \(a\) is the distance from the center to a vertex. Note: many US textbooks always use \(a\) for the longer half-axis and write a tall ellipse as \(\dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{a^{2}} = 1\); on this page, \(a^{2}\) is always the \(x^{2}\) denominator. |
| \(c\) | see | The distance from the center to a focus. The foci are at \((\pm c,\ 0)\) or \((0,\ \pm c)\). It is the third letter after \(a\) and \(b\). For an ellipse, \(c^{2} = a^{2} - b^{2}\); for a hyperbola, \(c^{2} = a^{2} + b^{2}\). |
| \(F,\ F'\) | F, F prime | Names of the foci, from the first letter of "focus". An ellipse or a hyperbola has two foci, and the second one is written \(F'\) (F prime). |
| \(P\) | pee | The name of a point on the curve, from the first letter of "point". It is used to state definitions, as in "for every point \(P\), the sum of the distances is constant". |
| \(PF\) | P F | The length of the segment joining the point \(P\) and the focus \(F\) (the distance from \(P\) to \(F\)). Writing the names of two points side by side for the length of a segment is the usual notation in geometry. |
| \(H\) | H | The name of the foot of the perpendicular from the point \(P\) on the parabola to the directrix (where it meets the directrix). \(H\) is a common letter for the foot of a perpendicular (said to come from "height"). The distance from \(P\) to the directrix is written \(PH\). |
| \(p\) | pee | For the parabola \(y^{2} = 4px\), the number that gives the distance from the vertex to the focus (a negative value gives the opposite direction). The focus is \((p,\ 0)\) and the directrix is \(x = -p\). The letter is said to come from "parameter". |
| \(e\) | e | The eccentricity, from the first letter of "eccentricity". A circle has \(e = 0\), an ellipse \(0 < e < 1\), a parabola \(e = 1\) and a hyperbola \(e > 1\). (It is not the number e ≈ 2.718.) |
| \(\pm\) | plus or minus | A sign that covers both the + case and the − case at once. The foci \((\pm 4,\ 0)\) are the two points \((4,\ 0)\) and \((-4,\ 0)\) written together. |
| \(\sqrt{\ }\) | square root | The sign for the square root (the positive number whose square is the number inside). It is used to get \(c = \sqrt{5}\) from \(c^{2} = 5\). \(\sqrt{5}\) is about 2.236. |
Terms
| conic section | A curve that is the edge you get when you slice a cone (like an ice cream cone) with a plane. Depending on the angle of the cut, the edge is a circle, an ellipse, a parabola or a hyperbola. They are taught in Precalculus. |
| second-degree curve | Another way to describe the conic sections: curves given by an equation of degree 2 in \(x\) and \(y\). Ellipses, hyperbolas, parabolas (and circles) are exactly these curves. |
| ellipse | The set of points whose distances to two foci add up to a constant. It looks like a circle squashed in one direction, and its standard form is \(\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1\). |
| hyperbola | The set of points whose distances to two foci differ by a constant. It is a pair of curves opening left and right (or up and down), and its standard form is \(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = \pm 1\). |
| parabola | The set of points at the same distance from a focus and a directrix. It is the shape of the path of a thrown ball. Its standard form is \(y^{2} = 4px\) (or \(x^{2} = 4py\)). |
| focus | A special point that determines the shape of a conic section (plural: foci). An ellipse or a hyperbola has two, and a parabola has one. Foci gather light and radio waves, and the word "focus" comes from the Latin for "hearth" or "fireplace". |
| directrix | The line used to define a parabola. Every point on the parabola is the same distance from the focus as from the directrix. The directrix of \(y^{2} = 4px\) is \(x = -p\). |
| asymptote | A line that a curve gets closer and closer to but never reaches. The hyperbola \(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = \pm 1\) has two asymptotes, \(y = \pm\dfrac{b}{a}x\). |
| eccentricity | One number \(e\) that tells how far a curve is from being a circle ("eccentric" comes from "off-center"). With circle \(e = 0\), ellipse \(0 < e < 1\), parabola \(e = 1\) and hyperbola \(e > 1\), this one number sorts all the conic sections. |
| major axis | The longest line segment across an ellipse through its center. For a wide ellipse its length is \(2a\), and the foci lie on the major axis. |
| minor axis | The shortest line segment across an ellipse through its center. For a wide ellipse its length is \(2b\), and it crosses the major axis at a right angle. |
| vertex | A point where the curve meets an axis of symmetry (plural: vertices). An ellipse has four (the ends of the major and minor axes; some textbooks call only the ends of the major axis vertices and the others co-vertices), a hyperbola has two, and a parabola has one (the origin). |
| standard form | The simplest form of the equation, with the center (for a parabola, the vertex) at the origin, such as \(\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1\). It makes the foci and vertices easy to read. |
| square root | A number whose square is the given number. Getting \(c = 4\) from \(c^{2} = 16\) is "taking the square root". When the square root is not a whole number, the exact way to write it is with the root sign, such as \(\sqrt{5}\). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to review these topics is the fastest way forward.
| Square roots (Grade 8 and Algebra 1) |
|
| The coordinate plane and the distance formula (Grades 6–8 and Geometry) |
|
| Graphs of quadratic functions (Algebra 1) |
|
| Equations of circles (Geometry and Algebra 2) |
|
| Fractions and ratios (Grades 5–7) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the foci and eccentricity of an ellipse (wide, a² > b²)
| x² denominator a² | 25 |
| y² denominator b² | 9 |
| c² = a² − b² | =B1-B2 |
| Distance to a focus c = √(c²) | =SQRT(B3) |
| Major axis length 2a | =2*SQRT(B1) |
| Minor axis length 2b | =2*SQRT(B2) |
| Eccentricity e = c/a | =B4/SQRT(B1) |
Table to find the foci and asymptotes of a hyperbola
| x² denominator a² | 9 |
| y² denominator b² | 16 |
| c² = a² + b² | =B1+B2 |
| Distance to a focus c = √(c²) | =SQRT(B3) |
| Slope of the asymptotes b/a | =SQRT(B2/B1) |
| Eccentricity e = c/a | =B4/SQRT(B1) |
Table to find the focus and directrix of a parabola (y² = cx)
| x coefficient c | 8 |
| p = c ÷ 4 | =B1/4 |
| Focus x-coordinate (the focus is (p, 0)) | =B2 |
| Directrix x = −p | =-B2 |
After pasting, the upper rows (denominators or coefficient) are your inputs and the lower rows are calculated automatically. SQRT gives the square root.
The first table is the ellipse x²/25 + y²/9 = 1: c² = 16, c = 4, major axis 10, minor axis 6 and eccentricity 0.8. For a tall ellipse (a² < b²), change the c² formula to "=B2-B1" and the eccentricity to "=B4/SQRT(B2)".
The second table is the hyperbola x²/9 − y²/16 = 1: c² = 25, c = 5, asymptote slope 1.333… (= 4/3) and eccentricity 1.666… (= 5/3).
The third table is the parabola y² = 8x: p = 2, focus (2, 0) and directrix x = −2. For x² = cy (opens up or down), read the focus as (0, p) and the directrix as y = −p.
Excel gives decimal answers, so use the calculator on this page when you need exact values such as √5.
The first table is the ellipse x²/25 + y²/9 = 1: c² = 16, c = 4, major axis 10, minor axis 6 and eccentricity 0.8. For a tall ellipse (a² < b²), change the c² formula to "=B2-B1" and the eccentricity to "=B4/SQRT(B2)".
The second table is the hyperbola x²/9 − y²/16 = 1: c² = 25, c = 5, asymptote slope 1.333… (= 4/3) and eccentricity 1.666… (= 5/3).
The third table is the parabola y² = 8x: p = 2, focus (2, 0) and directrix x = −2. For x² = cy (opens up or down), read the focus as (0, p) and the directrix as y = −p.
Excel gives decimal answers, so use the calculator on this page when you need exact values such as √5.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the foci and eccentricity of an ellipse (wide, a² > b²)
| x² denominator a² | 25 |
| y² denominator b² | 9 |
| c² = a² − b² | =B1-B2 |
| Distance to a focus c = √(c²) | =SQRT(B3) |
| Major axis length 2a | =2*SQRT(B1) |
| Minor axis length 2b | =2*SQRT(B2) |
| Eccentricity e = c/a | =B4/SQRT(B1) |
Table to find the foci and asymptotes of a hyperbola
| x² denominator a² | 9 |
| y² denominator b² | 16 |
| c² = a² + b² | =B1+B2 |
| Distance to a focus c = √(c²) | =SQRT(B3) |
| Slope of the asymptotes b/a | =SQRT(B2/B1) |
| Eccentricity e = c/a | =B4/SQRT(B1) |
Table to find the focus and directrix of a parabola (y² = cx)
| x coefficient c | 8 |
| p = c ÷ 4 | =B1/4 |
| Focus x-coordinate (the focus is (p, 0)) | =B2 |
| Directrix x = −p | =-B2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the denominators or the coefficient with your own numbers.
How to calculate it in Python
from fractions import Fraction
import math
# Denominators of the ellipse x²/a² + y²/b² = 1 (a fraction such as 3/4 can be written as Fraction(3, 4))
x_denominator = Fraction(25) # a²
y_denominator = Fraction(9) # b²
# Ellipse: c² = larger denominator − smaller denominator (for a hyperbola, add them instead)
c_squared = abs(x_denominator - y_denominator)
c = math.sqrt(c_squared)
semi_major = math.sqrt(max(x_denominator, y_denominator))
print(f"c² = {c_squared}")
print(f"c = √{c_squared} = {c}")
if x_denominator >= y_denominator:
print(f"Foci: (±{c}, 0)") # wide → foci on the x-axis
else:
print(f"Foci: (0, ±{c})") # tall → foci on the y-axis
print(f"Eccentricity e = c ÷ {semi_major} = {c / semi_major}")
Runs with the standard library only. The fractions module keeps the denominators as fractions, so everything up to c² is exact with no rounding error (only the square root, math.sqrt, gives a decimal). This example is the ellipse x²/25 + y²/9 = 1. Running it prints four lines: "c² = 16", "c = √16 = 4.0", "Foci: (±4.0, 0)" and "Eccentricity e = c ÷ 5.0 = 0.8". For the hyperbola x²/a² − y²/b² = 1, change the c_squared line to "x_denominator + y_denominator". For the parabola y² = cx, you only need p = c/4, which you can get directly, as in Fraction(8) / 4.
How to write it in LaTeX and other math languages (copy and paste)
Foci of an ellipse (\(\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1\), \(a > b > 0\))
c² = a² − b²
c^{2} = a^{2} - b^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>c</mi><mn>2</mn></msup>
<mo>=</mo>
<msup><mi>a</mi><mn>2</mn></msup>
<mo>−</mo>
<msup><mi>b</mi><mn>2</mn></msup>
</mrow>
</math>
c^2 = a^2 - b^2
c = Sqrt[a^2 - b^2]
c := sqrt(a^2 - b^2);
c = sqrt(a^2 - b^2);
c^2 = a^2 - b^2
Foci and asymptotes of a hyperbola (\(\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1\))
c² = a² + b²
c^{2} = a^{2} + b^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>c</mi><mn>2</mn></msup>
<mo>=</mo>
<msup><mi>a</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>b</mi><mn>2</mn></msup>
</mrow>
</math>
c^2 = a^2 + b^2
c = Sqrt[a^2 + b^2]
c := sqrt(a^2 + b^2);
c = sqrt(a^2 + b^2);
c^2 = a^2 + b^2
Focus and directrix of a parabola (\(y^{2} = 4px\))
y² = 4px
y^{2} = 4px
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>y</mi><mn>2</mn></msup>
<mo>=</mo>
<mn>4</mn><mi>p</mi><mi>x</mi>
</mrow>
</math>
y^2 = 4px
y^2 == 4 p x
y^2 = 4*p*x;
y^2 == 4*p*x
y^2 = 4px
Eccentricity \(e\) (a number that describes the shape of the curve)
e = c/a
e = \dfrac{c}{a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>e</mi>
<mo>=</mo>
<mfrac><mi>c</mi><mi>a</mi></mfrac>
</mrow>
</math>
e = c/a
e = c/a
e := c/a;
e = c/a;
e = c/a
How to have ChatGPT do the calculation
You are a math assistant for conic sections (Precalculus). Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the ellipse x²/9 + y²/4 = 1, show each of the following: 1. The value of c² = a² − b² 2. The distance c from the center to a focus (both as an exact value with √ and as a decimal) 3. The coordinates of the foci 4. The length of the major axis 2a and the length of the minor axis 2b 5. The eccentricity e = c/a (both as an exact value and as a decimal) In Python, use sympy (or fractions and math if sympy is not available), and get exact values such as √5 with sympy.sqrt. Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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