Hyperbola Calculator
This hyperbola calculator takes the equation in any form and returns the centre, a, b and c, the vertices, both foci, the eccentricity and both asymptotes — with the asymptote slope left exact rather than rounded.
Type x²/9 − y²/16 = 1 and you get centre (0, 0), a = 3, b = 4, c = 5, and the two asymptotes y = 4/3 x and y = −4/3 x. Not 1.333.
What the Hyperbola Calculator Is Working From
Pick two points and call them the foci. A hyperbola is every point whose two distances to them differ by the same amount — and that amount is 2a.
An ellipse adds those two distances; a hyperbola subtracts them. One word is the entire difference between the two chapters, and almost every formula that follows flips because of it.
The Positive Term Decides Which Way It Opens
Divide through until the right-hand side is 1, then look at which term carries the plus sign:
x² positive → the branches open left and right
y² positive → they open up and down
That is all there is to it, and it has nothing to do with which denominator is larger. Reading the direction from the size of the numbers is the ellipse habit carried across, and it is wrong here.
a Belongs to the Positive Term, Big or Small
On an ellipse, a is the larger of the two. On a hyperbola it is simply whichever denominator sits under the positive term.
In x²/4 − y²/9 = 1 the answer is a = 2 and b = 3. That looks wrong to anyone still carrying the ellipse rule, and it is one of the two mistakes that cost the most marks in this topic.
Add for c, Do Not Subtract
c² = a² + b²
This is the other one. There is a reason worth keeping instead of a rule to memorise: the foci of a hyperbola lie outside the curve, out past the vertices, so c has to come out bigger than a — and only adding can do that. On an ellipse the foci are inside, so it subtracts.
Check your answer against that every time. If c came out smaller than a, you have subtracted.
The Asymptotes Come Free with a and b
Two straight lines through the centre that the branches close in on and never reach:
y − k = ±(b/a)(x − h) when it opens left and right
y − k = ±(a/b)(x − h) when it opens up and down
The quickest way to draw them is the box: from the centre go a either way along the axis it opens on, and b either way across it. The two diagonals of that rectangle, extended, are the asymptotes — and the vertices sit on the middle of two of its sides.
Keep the Slope as a Surd
b/a is irrational whenever the ratio is not a perfect square, which is most of the time. For x²/3 − y²/2 = 1 the slope is √6/3, and there is no tidier way to write it.
Round it to 0.816 and you have a different line. The gap between the true asymptote and the rounded one grows the further out you draw, so a sketch built on the decimal drifts away from the branches it was meant to guide.
What the Eccentricity Says Here
e = c/a, and on a hyperbola it is always greater than 1 — because c is bigger than a. That single fact tells you the shape at a glance: just above 1 gives two narrow branches hugging their axis, and a large e gives branches that open out almost into straight lines.
An ellipse has e between 0 and 1, a parabola sits exactly at 1, and a hyperbola is everything above. The three conics are one family sorted by that number.
When It Is Not a Hyperbola
A hyperbola needs x² and y² with opposite signs. Same signs give a circle or an ellipse; only one squared term gives a parabola.
Two cases are worth naming. If the constant lands on zero, the two branches have closed up onto the pair of straight lines their asymptotes would have been — that is two crossing lines, not a hyperbola. And an equation with an xy term can still be one: xy = 1 is the plainest hyperbola there is, just turned, with the coordinate axes themselves as its asymptotes.
Check It on the Curve
Take any point on either branch and measure to both foci. The two distances must differ by 2a, every time, because that difference is the definition rather than a consequence of it.
It catches a wrong c, a swapped a and b, and the add-versus-subtract slip in a single measurement — which is more than re-reading your own working will do.
Using the Hyperbola Calculator
Put the equation in the box, however it is written, and tap the button. You get the centre, a, b and c, both vertices, both foci, the eccentricity, both axis lengths, both asymptotes and the latus rectum, a full working that completes the squares line by line, and a drawing of both branches with the asymptote box that produces them.
Hyperbola Calculator FAQ
How do you find the asymptotes of a hyperbola?
Is it a squared plus b squared for a hyperbola?
Which way does a hyperbola open?
Is a always bigger than b in a hyperbola?
Is xy = 1 a hyperbola?
Find the positive term, take a from it whatever its size, add for c — and keep the slope as a surd.