Coordinate Geometry (Straight Lines and Conic Sections)
Coordinate Geometry (Straight Lines and Conic Sections)
UPMSP tests equations, slopes, distances, and standard conic forms. A straight line can be written as y=mx+c, ax+by+c=0, or through a point with a given slope. Parallel lines have equal slopes; perpendicular slopes multiply to −1. For conics, identify the standard form before reading the centre, vertex, focus, directrix, or eccentricity.
Core rules
- Slope through (x1,y1) and (x2,y2): m=(y2−y1)/(x2−x1).
- Point-slope form: y−y1=m(x−x1).
- Distance from (x1,y1) to ax+by+c=0 is |ax1+by1+c|/√(a^2+b^2).
- Circle: (x−h)^2+(y−k)^2=r^2.
- Parabola: y^2=4ax has focus (a,0), directrix x=−a.
- Ellipse: x^2/a^2+y^2/b^2=1 with a>b has foci (±c,0), c^2=a^2−b^2, e=c/a.
- Hyperbola: x^2/a^2−y^2/b^2=1 has foci (±c,0), c^2=a^2+b^2.
Worked example
Find the line through (1,2) perpendicular to 3x+4y=7. The given line has slope −3/4, so the perpendicular slope is 4/3. Using point-slope form: y−2=(4/3)(x−1). Multiply by 3: 3y−6=4x−4, so 4x−3y+2=0. Substituting (1,2) gives 4−6+2=0, confirming the point lies on the line.
Traps
Dropping the absolute value in distance; using the reciprocal instead of the negative reciprocal for perpendicular lines; mixing a and b in ellipse formulas; using c^2=a^2−b^2 for a hyperbola; reading the vertex of y^2=4ax as (0,a).