←All concepts
Concept · Geometry & Algebra
Lines, circles, and distance
Practice this concept→
A Geometry & Algebra set, untimed
Named traps
9 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Circle positive-radius condition missedChecks coefficient equality but forgets the squared-radius must be positive (and no xy term).→
- Common chord line not derivedFails to subtract the two circle equations to get the radical/chord line.→
- Distance absolute value droppedConsiders only one sign case when two points are equidistant from a line, missing a solution.→
- Inclination-to-slope errorUses tan of the wrong angle, or confuses slope 1 with -1, from an inclination angle.→
- Perpendicular slope not reciprocalForgets the negative-reciprocal relationship for perpendicular lines.→
- Point-to-line distance errorErrs in the |Ax0+By0+C|/√A2+B2 formula (denominator or absolute value).→
- Reflection across wrong line/axisReflects a point across an axis instead of the given oblique line, or forgets the midpoint-on-line condition.→
- Tangent-circle radius from wrong axisTakes the radius from the wrong center coordinate when a circle is tangent to an axis.→
- Vertical-slope case missedOmits the vertical/undefined-slope case (or picks the complementary interval) for a line meeting a segment.→
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?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations