Conic Section Calculator

WHAT IT IS

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Type the three coefficients, then press =.

The squares completed, and the curve drawn

Any second-degree equation

Paste any second-degree equation, with or without an xy term. If it has one the curve is turned, and this page takes the turn out and hands you the straight equation — which every other conic page can then read. Degenerate cases are named too: a point, a line, a pair of lines, or nothing at all.

An xy term is welcome — that is what this page is for.

Tap a box, then use the keys

Every conic is Ax² + Bxy + Cy² + Dx + Ey + F = 0. Two numbers settle it: B² − 4AC says which family — negative an ellipse, zero a parabola, positive a hyperbola — and the 3 × 3 determinant says whether it is a real curve or has collapsed to a point, a line or nothing. A B that is not zero means the curve is turned, and both of those numbers survive the turn being taken out.

Examples — tap one and it solves

The working, step by step

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This conic section calculator takes any second-degree equation, tells you what it is, and — when it carries an xy term — takes the rotation out and hands you the straight equation back.

Paste 5x² + 4xy + 5y² = 9 and you get: an ellipse, turned, and the same curve written as 7x² + 3y² − 9 = 0 with no xy term left in it.

What the Conic Section Calculator Reads

Every conic is Ax² + Bxy + Cy² + Dx + Ey + F = 0. Six numbers, and two things made from them decide everything.

B² − 4AC gives the family: negative is an ellipse, zero a parabola, positive a hyperbola. The 3 × 3 determinant says whether it is a real curve at all, or has flattened into something simpler.

One Number Names the Family

The useful part of B² − 4AC is that it works whether or not the equation is turned. You can name the family of an equation you have no idea how to draw, before doing any algebra on it.

That is because a rotation cannot change it — and that same fact is what makes the rest of this page possible.

The Determinant Catches the Collapsed Ones

B² − 4AC alone cannot tell an ellipse from a single point, or a hyperbola from a pair of crossing lines. Both of those carry exactly the same family number as the real curve.

The 3 × 3 determinant can. When it comes out zero, the conic has flattened, and the family only tells you which way: an ellipse to a point or to nothing, a parabola to parallel lines, a hyperbola to two crossing lines.

An xy Term Means It Is Turned

If B is not zero, the curve's axes are not level with x and y. That is the whole meaning of the term, and it is why nothing from the standard chapter works: you cannot complete the square on an equation whose axes are at an angle.

Which is why almost every conic tool stops at "this is a rotated ellipse" and goes no further.

Taking the Turn Out Without Any Trigonometry

The textbook route runs through cot 2θ, then sinθ and cosθ, and drags a decimal through every step after that. It is not necessary.

Two quantities are unchanged by a rotation: A + C and B² − 4AC. If the straightened equation is A′x² + C′y² + … then A′ + C′ must equal A + C, and A′C′ must equal AC − B²/4. Two facts, two unknowns:

t² − (A + C)t + (AC − B²/4) = 0

The two roots of that are the new coefficients. No angle appears anywhere, so nothing is rounded.

Then Move to the Centre

A turned ellipse or hyperbola has a centre, found by solving 2Ax + By + D = 0 and Bx + 2Cy + E = 0. Shift there and the linear terms vanish, leaving A′x² + C′y² + F′ = 0.

A parabola has no centre, so the determinant does that job instead: it gives D′² = −4Δ/(A + C), the linear term of the straightened form. Different route, same exactness.

What the Angle Actually Is

tan 2θ = B/(A − C). When A and C are equal that denominator is zero and the turn is exactly 45° — which is why xy = 1 is a hyperbola tilted at 45°, with the axes themselves as its asymptotes.

For every other case the angle is genuinely irrational, and this page says so rather than printing a decimal and calling it the answer. You do not need it: the straightened equation comes out without it.

What to Do With the Straight Equation

Once the xy term is gone, the equation is the kind every chapter starts with. Complete the square, read the centre, the radii, the foci — all the ordinary machinery works again.

That is the point of the page. Copy the straightened equation into whichever conic tool you actually needed, and it will read it.

Using the Conic Section Calculator

Put the equation in the box, in any form, with or without an xy term. You get the family, both discriminants with what each one meant, whether the curve is real or collapsed, the turn taken out where there is one, the centre where there is one, and a drawing of what you typed next to the straightened version of it.

Conic Section Calculator FAQ

How do you identify a conic section from its equation?

Work out B squared minus 4AC. Negative gives the ellipse family, zero the parabola family, positive the hyperbola family. It works whether or not the equation has an xy term.

What does the xy term mean in a conic?

It means the curve is turned, so its axes are not level with x and y. Nothing from the standard chapter applies until the rotation is taken out.

How do you remove the xy term without trigonometry?

A plus C and B squared minus 4AC are both unchanged by a rotation. So the two new coefficients are the roots of t squared minus A plus C times t plus AC minus B squared over four, and no angle appears anywhere.

What is a degenerate conic?

One that has flattened into a point, a line, a pair of lines or nothing at all. The 3 by 3 determinant is zero exactly when that has happened.

Is xy = 1 a conic section?

Yes, a hyperbola turned through 45 degrees. Its asymptotes are the coordinate axes themselves.

Two numbers name it, and two invariants straighten it — no angle required.