Analytic geometry is the unity of algebra and geometry. It is often necessary to combine algebraic operation reasoning with geometric argumentation. When solving problems, we should make full use of the rigor of algebraic operation and the intuition of geometric argument. In particular, we should use some algebraic expressions to imagine the geometric meaning of some graphics, and use the properties of graphics to explain algebraic properties.

Parameter method

The (1) point parameter uses a point to set a point (permanent "active point") on the curve. Taking this point as a parameter, other related quantities are solved in turn, and then solved by formula. For example, a fixed point P and a station P(t, 0) on the X axis; A moving point p on the straight line x-2y+ 1=0. In addition to setting P(x 1, y 1), you can also set P(2y,-1, y 1) directly.

(2) Slope is a parameter.

When a straight line passes through a certain point P(x0, y0), let this straight line be y-y0=k(x-x0), that is, take k as the parameter, and then solve it in turn according to the proposition requirements.

(3) Angle parameters

When we study the problems about rotation, we always take a certain angle as the parameter, especially the fixed point problems on circles and ellipses.

substitution method

The "substitution method" mentioned here mainly refers to the substitution method of conditions in different orders. For example, for the proposition: "Given the condition P 1, P2 finds (or verifies) the target q", method 1 substitutes the condition P 1 into the condition P2, method 2 substitutes the condition P2 into the condition P 1, and method 3 substitutes the target q into to be determined. Different substitution methods often affect the difficulty of solving problems, so we should learn to analyze and choose simple substitution methods.