Foreword
Conic generally refers to an ellipse, hyperbola, parabola; but because circles and ellipses have close relatives, are closed curve, and when the two combined focus of the ellipse, the ellipse becomes a circle; are non-parabolic and hyperbolic closed curve, the difference between the two front and two on the large one.
Basics
- Straight \ (L \) and conic \ (C \) positional relationship
1, from a geometric point of view, the straight line \ (L \) and conic \ (C \) positional relationship can be divided into three categories: ① no common point; only one common point ②; ③ a common two distinct point;
2, the number of angles of view, can be solved by the method is determined by algebraic substitution method. Typically the linear \ (L \) equation \ (Ax of + By + C = 0 (A ^ 2 + B ^ 2 \ NEQ 0 \) , or \ (A \) , \ (B \) are not simultaneously \ (0 \) ) is substituted into the conic \ (C \) equation \ (F (x, y) = 0 \) , the erasing \ (Y \) (or \ (X \) ) to give about variable \ (X \) (or a variable \ (Y \) ) mono- equation (imitation quadratic), i.e., by the \ (\ left \ {\ begin {array} {l} {Ax + By + C = 0} \\ . F. {(X, Y) = 0} \} End {Array \ right \) , erasing \ (Y \) to give \ (AX + BX + C ^ 2 = 0 \) ;
(1) When \ (a \ neq 0 \) when a quadratic equation set \ (ax ^ 2 + bx + c = 0 \) discriminant is \ (\ of Delta \) , there
\ (\ Of Delta> 0 \) \ (\ Leftrightarrow \) straight \ (L \) conic \ (C \) intersect at different points;
\ (\ Of Delta = 0 \) \ (\ Leftrightarrow \) straight \ (L \) conic \ (C \) is tangent;
\ (\ Of Delta <0 \) \ (\ Leftrightarrow \) straight \ (L \) conic \ (C \) phase from no common point;
(2) When \ (A = 0 \) , \ (B \ NEQ 0 \) , i.e. to obtain a linear equation, the linear \ (L \) conic, and only one intersection point; in this case
If \ (C \) is hyperbolic, the straight line \ (L \) hyperbolic \ (C \) positional relationship between the asymptotes are parallel;
If \ (C \) is a parabola, the straight line \ (L \) parabola \ (C \) position of the symmetry axis is parallel or coincident;
Typical Example Analysis
Method 1: From the number of analog \ (y = kx + 1 \ ) constant through the point \ ((0,1) \) method ideas, so that \ (Y = 0 \) , to give \ (x ^ 2 = 1 \) , so that the above-described constant curve over point \ ((\ PM 1,0) \) ;