The circle

Where the conics come from

• A conic section is the curve formed when a plane slices a double cone.
• The angle of the slice decides which curve you get:
  • Perpendicular to the axis → circle
  • Tilted, but less steeply than the slope of the cone → ellipse
  • Exactly parallel to the slope of the cone → parabola
  • Steeper than the slope, cutting both halves of the cone → hyperbola
• The circle is a special case of the ellipse, with both axes equal — which is why their equations are so similar.

The equation of a circle

• A circle is the set of points at a fixed distance $r$ (the radius) from a fixed point $(h, k)$ (the centre).
• Applying the distance formula to that definition gives the standard form:

$$(x - h)^2 + (y - k)^2 = r^2$$

• Each part:
  • $(h, k)$ — the centre
  • $r$ — the radius, always taken positive
• The signs are subtracted inside the brackets. $(x + 3)^2$ means $h = -3$, not $+3$. This one sign error accounts for most wrong centres.
• Centred at the origin the equation reduces to $x^2 + y^2 = r^2$.

Recognising a circle from the general form

• Expanding the standard form gives the general form:

$$x^2 + y^2 + Dx + Ey + F = 0$$

• Two features identify a circle in this form:
  • Both $x^2$ and $y^2$ appear, and
  • their coefficients are equal (and there is no $xy$ term).
• If the coefficients are equal but not 1, divide the whole equation through first. $2x^2 + 2y^2 - 8x = 10$ becomes $x^2 + y^2 - 4x = 5$.

Completing the square

• This is the core technique of the entire standard. It converts the general form into the standard form.
• The steps:
  • Group the $x$ terms together and the $y$ terms together, and move the constant to the right.
  • Halve the coefficient of $x$, square it, and add it to both sides. Repeat for $y$.
  • Factorise each group into a perfect square.
  • Read off the centre and radius.
• Worked through — $x^2 + y^2 - 6x + 8y - 11 = 0$:
  • Group: $(x^2 - 6x) + (y^2 + 8y) = 11$
  • Half of $-6$ is $-3$, squared is $9$; half of $8$ is $4$, squared is $16$:
  • $$(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16$$
  • $$(x - 3)^2 + (y + 4)^2 = 36$$
  • Centre $(3, -4)$, radius $6$
• Whatever you add on the left must be added on the right. Forgetting that changes the radius but not the centre, so the answer looks plausible.

When it is not a circle

• After completing the square, the right-hand side may not be a positive number:
  • Positive → a genuine circle of radius $\sqrt{\text{RHS}}$
  • Zero → a single point, sometimes called a degenerate circle
  • Negative → no real points at all — the equation has no graph
• Report the degenerate cases rather than taking the square root of a negative.

Intersections with a line

• Substitute the line into the circle and solve the resulting quadratic.
• The discriminant tells you the geometry without solving:

• The alternative check is often faster: compare the perpendicular distance from the centre to the line with the radius. Less than $r$ → cuts; equal → tangent; greater → misses.

Properties worth knowing

• The tangent at any point is perpendicular to the radius drawn to that point. This single fact solves most circle-tangent questions without any calculus.
• The perpendicular from the centre to a chord bisects the chord.
• A circle is the only conic with constant curvature — every point looks the same, which is why circles have no foci to distinguish and no eccentricity to speak of (formally, $e = 0$).