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Concept · Geometry & Algebra
Parabola
definition (focus/directrix), standard forms, focal distance to x-coordinate, weighted-distance minimization via directrix
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Key formulas
9- Slope from two pointsk = (y2 − y1)/(x2 − x1)vertical line slope undefinedslope from inclination θ: k = tan θ
- Line formspoint-slope y−y1=k(x−x1)slope-intercept y=kx+bintercept x/a+y/b=1general Ax+By+C=0
- Parallel: k1=k2 (or A1B2=A2B1)Perpendicular: k1·k2=−1 (or A1A2+B1B2=0)
- Point-to-line distanced = |Ax0+By0+C| / √A2+B2
- Circle standard(x−a)2+(y−b)2=r2general x2+y2+Dx+Ey+F=0, center (−D/2,−E/2), r=√D2+E2−4F/2 (need D2+E2−4F>0)
- Ellipsex2/a2 + y2/b2 = 1 (a>b>0)a2 = b2 + c2|PF1|+|PF2| = 2ae = c/a < 1
- Hyperbolax2/a2 − y2/b2 = 1c2 = a2 + b2asymptotes y = ±(b/a)x||PF1|−|PF2|| = 2ae = c/a > 1
- Parabolay2 = 2px (focus (p/2,0), directrix x=−p/2)e = 1distance to focus = distance to directrix
- Common chord of two circlessubtract the two general equations to get the chord (radical) line
Shaky underneath?
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FoundationsPrecalculus bridgeAlgebra 2 foundations