Two Dimensional Analytical Geometry II
Two Dimensional Analytical Geometry II tests circles, parabolas, ellipses and hyperbolas: standard equations, eccentricity, foci, directrices, tangents, normals and parametric forms. The exam often gives a conic and asks for a tangent or a geometric property.
Standard forms
Circle: (x−h)²+(y−k)²=r². Parabola: y²=4ax, focus (a,0), directrix x=−a; tangent at (x1,y1) is yy1=2a(x+x1). Ellipse: x²/a²+y²/b²=1 with a>b, e=√(1−b²/a²), foci (±ae,0), directrices x=±a/e; tangent at (x1,y1) is xx1/a²+yy1/b²=1. Hyperbola: x²/a²−y²/b²=1, e=√(1+b²/a²), foci (±ae,0); tangent at (x1,y1) is xx1/a²−yy1/b²=1. Parametric point on the ellipse is (a cosθ,b sinθ); on the parabola it is (at²,2at).
Worked example
For x²/25+y²/9=1, a=5, b=3, so e=√(1−9/25)=4/5. Foci are (±4,0), directrices x=±25/4, and the tangent at (5 cosθ,3 sinθ) is x cosθ/5 + y sinθ/3=1.
Traps
Do not assume the larger denominator is always under x²; identify the major axis first. The hyperbola eccentricity uses a plus sign inside the square root, unlike the ellipse. The tangent formula at a point on the conic is not the same as the slope form. For the parabola, the tangent yy1=2a(x+x1) uses 2a, not a.