Implicitly Defined Curves & Conic Sections

You’ll be able to

Distinguish an implicitly defined relation from a function
Identify a conic section from its equation and put it in standard form
Explain why an implicit equation may define no function at all

An equation in x and y need not define a function

x² + y² = 25 describes a circle. It is a perfectly good relation between x and y, but it is not a function: x = 3 gives both y = 4 and y = −4. Such an equation is implicitly defined — it constrains the pair (x, y) without expressing one as a formula in the other. Many implicit curves can be split into two or more functions (here y = √(25 − x²) and y = −√(25 − x²)), but the curve as a whole fails the vertical line test.

The four conic sections in standard form

circle: (x − h)² + (y − k)² = r² · ellipse: (x − h)²/a² + (y − k)²/b² = 1 · parabola: (y − k) = a(x − h)² · hyperbola: (x − h)²/a² − (y − k)²/b² = 1

The distinguishing feature is how the squared terms combine: both positive and equal coefficients gives a circle, both positive and unequal gives an ellipse, opposite signs gives a hyperbola, and only one squared term gives a parabola.

Completing the square recovers the center

A conic given as x² + y² − 6x + 4y − 12 = 0 hides its center. Group the x terms and the y terms, complete the square on each, and the standard form appears. The constants you add to complete each square must be added to the right side too — that bookkeeping is where the arithmetic usually goes wrong, and it is worth writing out rather than doing mentally.

Worked example

Put x² + y² − 6x + 4y − 12 = 0 in standard form and identify the curve.

1.Group: (x² − 6x) + (y² + 4y) = 12.
2.Complete the square in x: half of −6 is −3, and (−3)² = 9. Add 9 to both sides.
3.Complete the square in y: half of 4 is 2, and 2² = 4. Add 4 to both sides.
4.(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 = 25.
5.(x − 3)² + (y + 2)² = 25.

Answer: A circle centered at (3, −2) with radius 5. Both squared terms are present with the same positive coefficient, which is the signature of a circle; unequal positive coefficients would have made it an ellipse.

Checkpoint

Which curve does 4x² − 9y² = 36 describe?

Watch out

Check the right-hand side after completing the square. If it comes out negative for a sum of squares, there are no real points at all — the equation describes the empty set. If it comes out exactly zero, the "circle" is a single point.

Slicing a cone

The name "conic section" is literal: each curve is the intersection of a plane with a double cone. A plane perpendicular to the axis cuts a circle. Tilt it and you get an ellipse. Tilt it until it is parallel to one side of the cone and the closed curve opens up into a parabola. Tilt further, so the plane meets both halves of the double cone, and you get the two branches of a hyperbola. One family of shapes, distinguished only by angle.

Worked example

Does 9x² + 4y² = 36 define y as a function of x? Split it into functions if possible.

1.Divide by 36: x²/4 + y²/9 = 1, an ellipse with x-radius 2 and y-radius 3.
2.Solve for y: 4y² = 36 − 9x², so y² = 9 − 9x²/4 and y = ±3√(1 − x²/4).
3.The ± means two y values for each x strictly between −2 and 2, so the relation is not a function.
4.It splits into the upper half y = 3√(1 − x²/4) and the lower half y = −3√(1 − x²/4).
5.Each half is a function on the domain [−2, 2].

Answer: No — the ellipse fails the vertical line test. It splits into two functions, the upper and lower semi-ellipses, each with domain [−2, 2] and joined at the endpoints (±2, 0) where the two halves meet.

Checkpoint

After completing the square, an equation becomes (x + 1)² + (y − 4)² = −9. What does it describe?

Answer the 2 checkpoints as you read.