source: ncert
topic: geometry
class_level: jee_advanced
chapter: advanced_coord_geometry
difficulty: advanced
JEE Advanced | Topic 3: Advanced Coordinate Geometry PAIR OF STRAIGHT LINES: ax^2+2hxy+by^2=0 represents pair of lines through origin. Slopes m1,m2: m1+m2=-2h/b, m1m2=a/b. Angle between lines: tan(theta)=2sqrt(h^2-ab)/(a+b). Lines coincident: h^2=ab. Lines perpendicular: a+b=0. Combined equation of lines through origin with slopes m1,m2: b(y-m1x)(y-m2x)=0. General second degree: ax^2+2hxy+by^2+2gx+2fy+c=0. Represents pair of lines if det([[a,h,g],[h,b,f],[g,f,c]])=0. CIRCLES: Radical axis of two circles: locus of points with equal power w.r.t. both circles. Power of point P w.r.t. circle: PT^2=S1 where S1=x1^2+y1^2+2gx1+2fy1+c. Radical axis: S1-S2=0 (subtract equations). Coaxial circles: family of circles with common radical axis. CONICS - ADVANCED: Chord of contact from point (x1,y1) to circle x^2+y^2=a^2: xx1+yy1=a^2. Chord of contact to ellipse x^2/a^2+y^2/b^2=1: xx1/a^2+yy1/b^2=1. Parametric forms: Ellipse (acos t, bsin t). Parabola (at^2, 2at). Hyperbola (asec t, btan t). Normal at parametric point t on parabola y^2=4ax: y=-tx+2at+at^3. Condition for line y=mx+c to be tangent to y^2=4ax: c=a/m. LOCUS PROBLEMS: Set up coordinates, use given conditions, eliminate parameter, simplify to standard form. SOLVED EXAMPLES: Example 1: 3x^2+10xy+8y^2=0. Find angle between lines. a=3, 2h=10 so h=5, b=8. tan(theta)=2sqrt(25-24)/(3+8)=2/11. theta=tan^(-1)(2/11). Example 2: Chord of contact from (1,2) to circle x^2+y^2=25. x(1)+y(2)=25 → x+2y=25. Example 3: If z=x+iy and |z-2|=2|z-1|, find locus. |(x-2)+iy|=2|(x-1)+iy|. (x-2)^2+y^2=4[(x-1)^2+y^2]. x^2-4x+4+y^2=4x^2-8x+4+4y^2. 3x^2+3y^2-4x=0. x^2+y^2-4x/3=0. Circle. COMMON MISTAKES: Pair of lines: 2hxy (coefficient of xy is 2h, not h). Radical axis: subtract S1 and S2 equations directly (coefficients of x^2,y^2 must be 1 first). Tangent condition: c=a/m for parabola y^2=4ax (from OUTSIDE).