Formula Sheet — Conic Sections

Card 1: Circle

Fact Statement
Definition points at fixed distance rr from centre (h,k)(h, k)
Standard equation (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
General form x2+y2+2gx+2fy+c=0x^2 + y^2 + 2gx + 2fy + c = 0: centre (−g,−f)(-g, -f), radius g2+f2−c\sqrt{g^2 + f^2 - c}
Point position S1<0S_1 < 0 inside, =0= 0 on, >0> 0 outside

Card 2: Parabola (all with vertex at origin)

Equation Focus Directrix Latus rectum
y2=4axy^2 = 4ax (a,0)(a, 0) x=−ax = -a 4a4a
y2=−4axy^2 = -4ax (−a,0)(-a, 0) x=ax = a 4a4a
x2=4ayx^2 = 4ay (0,a)(0, a) y=−ay = -a 4a4a
x2=−4ayx^2 = -4ay (0,−a)(0, -a) y=ay = a 4a4a

Focal distance of (x1,y1)(x_1, y_1) on y2=4axy^2 = 4ax: x1+ax_1 + a.

Card 3: Ellipse (a>ba > b, c2=a2−b2c^2 = a^2 - b^2, e=ca<1e = \frac{c}{a} < 1)

Equation Foci Vertices Latus rectum
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (±c,0)(\pm c, 0) (±a,0)(\pm a, 0) 2b2a\frac{2b^2}{a}
x2b2+y2a2=1\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 (0,±c)(0, \pm c) (0,±a)(0, \pm a) 2b2a\frac{2b^2}{a}

Sum of focal distances =2a= 2a; focal distances a±ex1a \pm ex_1; larger denominator names the major axis.

Card 4: Hyperbola (c2=a2+b2c^2 = a^2 + b^2, e=ca≥1e = \frac{c}{a} \ge 1)

Equation Foci Vertices Asymptotes
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (±c,0)(\pm c, 0) (±a,0)(\pm a, 0) y=±baxy = \pm\frac{b}{a}x
y2a2−x2b2=1\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 (0,±c)(0, \pm c) (0,±a)(0, \pm a) y=±abxy = \pm\frac{a}{b}x

Difference of focal distances =2a= 2a; latus rectum 2b2a\frac{2b^2}{a}; the positive term names the transverse axis; a=ba = b gives the equilateral hyperbola with e=2e = \sqrt{2}.

Card 5: Tangency of y=mx+cy = mx + c (JEE Corner)

Conic Condition
x2+y2=r2x^2 + y^2 = r^2 c2=r2(1+m2)c^2 = r^2(1 + m^2)
y2=4axy^2 = 4ax c=amc = \frac{a}{m}
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 c2=a2m2+b2c^2 = a^2m^2 + b^2
x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 c2=a2m2−b2c^2 = a^2m^2 - b^2

Last-Minute Mistake Checklist

  1. (x+4)2(x + 4)^2 means h=−4h = -4 — read centres with the "xx minus hh" rule; sign slips on the centre are the top circle error.
  2. Divide first: 2x2+2y2−x=02x^2 + 2y^2 - x = 0 must become x2+y2−x2=0x^2 + y^2 - \frac{x}{2} = 0 before completing the square; similarly 9x2+4y2=369x^2 + 4y^2 = 36 must be divided by 36 before reading aa and bb.
  3. Ellipse vs hyperbola relation: c2=a2−b2c^2 = a^2 - b^2 (ellipse) but c2=a2+b2c^2 = a^2 + b^2 (hyperbola). Anchor: the hyperbola's foci lie beyond its vertices.
  4. Which axis is major/transverse: ellipse — the larger denominator; hyperbola — the positive term (denominator size is irrelevant).
  5. Latus rectum is 4a4a for the parabola but 2b2a\frac{2b^2}{a} for ellipse and hyperbola — don't transplant 4a4a across conics.
  6. Orientation before substitution: fitting a parabola through a point, check which quadrant the point is in first — the wrong form gives a negative aa.
  7. Eccentricity ranges: circle e=0e = 0, ellipse 0<e<10 < e < 1, parabola e=1e = 1, hyperbola e>1e > 1. An answer violating the range is self-diagnosed as wrong.
  8. Focal distance of a parabola point is x1+ax_1 + a — no distance formula needed; using (x1−a)2+y12\sqrt{(x_1-a)^2 + y_1^2} invites arithmetic slips (it gives the same answer, slower).
  9. Tangency sign pattern: +b2+b^2 for the ellipse, −b2-b^2 for the hyperbola; for the circle use distance-from-centre == radius.
  10. Real circle check: g2+f2−c>0g^2 + f^2 - c > 0 — if a parameter question yields a negative radius squared, the "circle" does not exist for that parameter.

How this chapter flows on: the next chapter lifts coordinates into three dimensions, and Class 12 calculus returns to these curves for tangents, normals, areas and maxima — with every standard equation here assumed known.