ConceptGeometry & Algebra

Key formulas

Slope from two points: k = (y2 − y1)/(x2 − x1); vertical line slope undefined; slope from inclination θ: k = tan θ
Line forms: point-slope y−y1=k(x−x1); slope-intercept y=kx+b; intercept x/a+y/b=1; general Ax+By+C=0
Parallel: k1=k2 (or A1B2=A2B1); Perpendicular: k1·k2=−1 (or A1A2+B1B2=0)
Point-to-line distance: d = |Ax0+By0+C| / √A²+B²
Circle standard: (x−a)²+(y−b)²=r²; general x²+y²+Dx+Ey+F=0, center (−D/2,−E/2), r=√D²+E²−4F/2 (need D²+E²−4F>0)
Ellipse: x²/a² + y²/b² = 1 (a>b>0); a² = b² + c²; |PF1|+|PF2| = 2a; e = c/a < 1
Hyperbola: x²/a² − y²/b² = 1; c² = a² + b²; asymptotes y = ±(b/a)x; ||PF1|−|PF2|| = 2a; e = c/a > 1
Parabola: y² = 2px (focus (p/2,0), directrix x=−p/2); e = 1; distance to focus = distance to directrix
Common chord of two circles: subtract the two general equations to get the chord (radical) line

If the concept underneath still feels loose, patch it here before returning to CSCA questions.

Lines, circles, and distance

9 ways this goes wrong. Each one is a named misconception the questions are built to catch.