Pre-Calculus Conic sections

Identifying conics from general form ax² + by² + cx + dy + e = 0

Q: How do you identify a conic from its general form?

Compare the coefficients of x squared and y squared: equal is a circle, same sign an ellipse, opposite signs a hyperbola, one missing a parabola.

Q: What makes a conic a parabola in general form?

Only one of x squared or y squared appears; the other squared term is missing.

Q: How do you tell an ellipse from a hyperbola?

Same sign on the squared coefficients gives an ellipse; opposite signs give a hyperbola.

20 practice questions 0 video lessons Theory + worked examples

Identifying Conics from General Form

Texas Precalculus (TEKS) • Standard P.3(F) • Conic Sections

Identifying Conics from General Form is a topic in Conic Sections in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.3(F), which requires students to determine the conic section represented by a second-degree equation.

Identifying a conic from \(Ax^2+Cy^2+Dx+Ey+F=0\) compares \(A\) and \(C\), then completes the square to reach standard form.

Practice 20 questions

Practice questions

Every question with a fully worked solution.

Start practising

Theory

A conic in general form (no \(xy\) term) is

\[Ax^2+Cy^2+Dx+Ey+F=0.\]

You can classify it just from \(A\) and \(C\):

Circle: \(A=C\) (and same sign).
Ellipse: \(A\) and \(C\) same sign but \(A\neq C\).
Hyperbola: \(A\) and \(C\) opposite signs.
Parabola: only one of \(x^2,y^2\) appears.

To find the center, radius, or axes, complete the square and rewrite in standard form.

Classify first, then complete the square. The coefficient test is instant; the algebra just fills in the details.

Classifying from the coefficients \(A\) and \(C\).

Then complete the square to reach standard form.

The general form and the classification test:

\[Ax^2+Cy^2+Dx+Ey+F=0\]

classify a conic from the coefficients A and C of the squared terms

Same sign \(\to\) ellipse/circle; opposite signs \(\to\) hyperbola; one square missing \(\to\) parabola.

How to identify and rewrite a conic

Compare \(A\) and \(C\) to classify the conic.
Group the \(x\)-terms and \(y\)-terms.
Complete the square in each variable.
Divide to reach standard form and read off center, radius, or axes.

Example 1 — Classify by coefficients

Classify \(4x^2+9y^2-16x+18y-11=0\).

Solution

Both squares present; coefficients \(4\) and \(9\) are the same sign but unequal.

A=4,\ C=9

\(\Rightarrow\)

\text{ellipse}

same sign, unequal coefficients, so it is an ellipse

Example 2 — A hyperbola

Classify \(x^2-4y^2+2x-8=0\).

Solution

The \(x^2\) and \(y^2\) coefficients have opposite signs.

A=1,\ C=-4

\(\Rightarrow\)

\text{hyperbola}

opposite signs, so it is a hyperbola

Example 3 — A parabola

Classify \(y^2-6x+4y+1=0\).

Solution

Only \(y\) is squared (no \(x^2\) term).

\text{one square}

\(\Rightarrow\)

\text{parabola}

only one variable squared, so it is a parabola

Example 4 — Complete the square

Put \(x^2+y^2-6x+4y+9=0\) into standard form.

Solution

Group and complete the square in \(x\) and \(y\).

(x^2-6x)+(y^2+4y)	\(=\)	-9
(x-3)^2+(y+2)^2	\(=\)	-9+9+4=4

A circle of radius 2 centered at \((3,-2)\).

a circle of radius 2 centered at 3 comma negative 2

Common pitfalls

Classify from the squared-term coefficients. The linear terms don't change the type.

Balance the equation when completing the square. Add the same amount to both sides.

One missing square means parabola, regardless of the other coefficients.

Frequently asked questions

How do you identify a conic from its general form?

Compare the coefficients \(A\) and \(C\) of \(x^2\) and \(y^2\): equal is a circle, same sign an ellipse, opposite signs a hyperbola, one missing a parabola.

Why complete the square?

To turn the general form into standard form, which reveals the center, radius, or axes of the conic.

What makes a conic a parabola in general form?

Only one of \(x^2\) or \(y^2\) appears; the other squared term is missing.

How do you tell an ellipse from a hyperbola?

Same sign on \(A\) and \(C\) gives an ellipse; opposite signs give a hyperbola.

← Previous subtopic

Locus definitions of conic sections

Next subtopic →

This is the last subtopic