Mathematics

Conic Sections

Circle, ellipse, parabola, and hyperbola. General and standard equations, calculation of foci, vertices, and asymptotes.

It is the locus of points P(x,y)P(x, y) that are equidistant from a fixed point called the center C(cx,cy)C(c_x, c_y).
Circle with center C and radius r
(x−cx)2+(y−cy)2=r2(x - c_x)^2 + (y - c_y)^2 = r^2
General equation:
x2+y2+Ax+By+C=0x^2 + y^2 + Ax + By + C = 0
(cx,cy)=(−A2,−B2)(c_x, c_y) = \left( -\frac{A}{2}, -\frac{B}{2} \right)
r2=(A2)2+(B2)2−Cr^2 = \left( \frac{A}{2} \right)^2 + \left( \frac{B}{2} \right)^2 - C
Standard (Canonical) equation of a circle centered at the origin C(0,0)C(0,0):
x2+y2=r2x^2 + y^2 = r^2
Power of a point P(x0,y0)P(x_0, y_0) with respect to a circle
If dd is the distance from the point to the center, the power is Pot=d2−r2Pot = d^2 - r^2.
Alternatively, evaluate the point in the general equation:
Pot=x02+y02+Ax0+By0+CPot = x_0^2 + y_0^2 + A x_0 + B y_0 + C
  • If Pot>0Pot > 0, the point is outside the circle.
  • If Pot=0Pot = 0, the point lies on the circle.
  • If Pot<0Pot < 0, the point is inside the circle.
Relative position of a line and a circle
Set up the system of equations for both:
{x2+y2+Ax+By+C=0y=mx+n\begin{cases} x^2 + y^2 + Ax + By + C = 0 \\ y = mx + n \end{cases}
  • 1 solution, the line is tangent.
  • 2 solutions, the line is secant.
  • If the system has no solution (0 sol.), the line is outside the circle.
Tangent and normal lines to a circle at a point (x0,y0)(x_0, y_0)
Tangent line and normal line to the circle at a point
The tangent is perpendicular to the radius:
y−y0=−x0−cxy0−cy(x−x0)y - y_0 = -\frac{x_0 - c_x}{y_0 - c_y}(x - x_0)

The normal is parallel to the radius:
y−y0=y0−cyx0−cx(x−x0)y - y_0 = \frac{y_0 - c_y}{x_0 - c_x}(x - x_0)
It is the locus of points P(x,y)P(x,y) such that the sum of their distances to two fixed points (called foci) is constant: PF+PF′=2aPF + PF' = 2a
Elements of the ellipse: foci, semi-axes, and focal distance
  • aa: major semi-axis
  • bb: minor semi-axis
  • cc: focal distance from center
a2=b2+c2a^2 = b^2 + c^2
Eccentricity: e=ca\quad e = \frac{c}{a}
General equation: Ax2+By2+Cx+Dy+E=0Ax^2 + By^2 + Cx + Dy + E = 0 \quad (where A and B have the same sign)
Standard equation of an ellipse centered at the origin (0,0)(0,0)
Horizontal Ellipse
Vertical Ellipse
Horizontal ellipse centered at the origin
Vertical ellipse centered at the origin
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
Standard equation of an ellipse centered at a point C(cx,cy)C(c_x, c_y)
Horizontal

(x−cx)2a2+(y−cy)2b2=1\frac{(x - c_x)^2}{a^2} + \frac{(y - c_y)^2}{b^2} = 1
Vertical

(x−cx)2b2+(y−cy)2a2=1\frac{(x - c_x)^2}{b^2} + \frac{(y - c_y)^2}{a^2} = 1
It is the locus of points in a plane that are equidistant from a fixed point FF (focus) and a fixed line dd (directrix).
Elements of the parabola: focus, directrix, and vertex
Where pp is the distance from vertex to focus or vertex to directrix:
p=d(F,V)=d(V,d)p = d(F, V) = d(V, d)

General equation:
Ax2+By2+Cx+Dy+E=0Ax^2 + By^2 + Cx + Dy + E = 0
(where A = 0 or B = 0)
Standard equation of a parabola with vertex at (vx,vy)(v_x, v_y)
Parabola opening upwards
Opens upwards
(x−vx)2=4p(y−vy)(x - v_x)^2 = 4p(y - v_y)
Focus: (vx,vy+p)(v_x, v_y + p)
Directrix: y=vy−py = v_y - p
Parabola opening to the right
Opens to the right
(y−vy)2=4p(x−vx)(y - v_y)^2 = 4p(x - v_x)
Focus: (vx+p,vy)(v_x + p, v_y)
Directrix: x=vx−px = v_x - p
Parabola opening downwards
Opens downwards
(x−vx)2=−4p(y−vy)(x - v_x)^2 = -4p(y - v_y)
Focus: (vx,vy−p)(v_x, v_y - p)
Directrix: y=vy+py = v_y + p
Parabola opening to the left
Opens to the left
(y−vy)2=−4p(x−vx)(y - v_y)^2 = -4p(x - v_x)
Focus: (vx−p,vy)(v_x - p, v_y)
Directrix: x=vx+px = v_x + p
It is the locus of points P(x,y)P(x,y) such that the absolute difference of their distances to two fixed foci is constant: ∣PF−PF′∣=2a|PF - PF'| = 2a
Elements of the hyperbola: semi-axes, foci, and asymptotes
  • aa: transverse semi-axis
  • bb: conjugate semi-axis
  • cc: focal distance from center
c2=a2+b2c^2 = a^2 + b^2
Eccentricity: e=ca\quad e = \frac{c}{a}
General equation: Ax2+By2+Cx+Dy+E=0Ax^2 + By^2 + Cx + Dy + E = 0 \quad (where A and B have opposite signs)
Standard equation of a hyperbola centered at the origin (0,0)(0,0)
Horizontal transverse axis
Vertical transverse axis
Horizontal hyperbola centered at the origin
Vertical hyperbola centered at the origin
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Asymptotes: y=±baxy = \pm \frac{b}{a}x
y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

Asymptotes: y=±abxy = \pm \frac{a}{b}x
Equation of an equilateral (rectangular) hyperbola (a=ba = b):
x2−y2=a2x^2 - y^2 = a^2