Solving skills of conic curve in high school mathematics 1. The strategy of making full use of geometric figures
The research object of analytic geometry is geometric figures and their properties, so when dealing with analytic geometry problems, besides using algebraic equations, fully mining geometric conditions and combining plane geometry knowledge can often reduce the amount of calculation.
Example: Let a straight line 3x+4y+m=0 and a circle x+y+x-2y=0 intersect at two points P and Q, and O is the coordinate origin. If OP? OQ, find the value of m
2. The strategy of making full use of Vieta's theorem.
We often set the coordinates of the end of a chord, but we can't find it. But combined with Vieta's theorem. This method is often used for slope, midpoint and other problems.
Example: It is known that the ellipse with the center of the circle at the origin O and the focus on the Y axis intersects with the straight line y=x+ 1 at the points P and Q, OP? OQ, |PQ|=, find this elliptic equation.
3. The strategy of making full use of curve equation
Example: Find the equation C:x+y-4x+2y=0, C:x+y-2y-4=0, and the center of the circle is on the straight line L: 2x+4Y- 1 = 0.
4. The strategy of making full use of elliptic parameter equation.
The parametric equation of ellipse involves sine and cosine. By using the boundedness of sine and cosine, the related problems of finding the maximum value can be solved. This is what we often call triangle replacement.
Example: p is the moving point on the ellipse += 1, a is the right end point of the major axis, and b is the upper end point of the minor axis. Find the maximum area of quadrilateral OAPB and the coordinates of point P at this time.
5. Several simple calculation strategies of line segment length
(1) Make full use of existing achievements and reduce the calculation process.
Find the chord length AB of the intersection of a straight line and a conic curve: substitute the straight line equation y=kx+b into the conic curve equation, and get the equation ax+bx+c=0. Let the two roots of the equation be x, x and the discriminant be △, then |AB|=? |x-x|=？ If the conclusion is directly used, the calculation process of formulas and prescriptions can be reduced.
Example: Find the length of line segment AB cut by ellipse X+4Y =1= 0.
(2) Combined with the special position relationship of graphics, the amount of calculation is reduced.
When calculating the chord length of the focus of the conic, because the definition of the conic involves the focus, using the definition of the conic in combination with graphics can avoid complicated operations.
Example: F and F are the two focuses of an ellipse += 1, and AB is the chord passing through F. If |AB|=8, find the value of |FA|+|FB|.
(3) Using the definition of conic curve, the distance from the focus is converted into the distance from the directrix.
Example: point a (3,2) is a fixed point, point f is the focus of parabola y=4x, and point p moves on parabola y=4x. If |PA|+|PF| gets the minimum value, find the coordinates of point P. ..
High school mathematics conic problem 1. Midpoint chord problem
For the chord midpoint problem with slope, the point difference method is often used: let two points on the curve be (x, y) and (x, y), substitute them into the equation, then subtract the two equations, and then use the midpoint relationship and slope formula to eliminate the four parameters.
Example: Given a hyperbola x-= 1, a straight line passing through A (2, 1) intersects with the hyperbola at two points P and P, and the trajectory equation of the midpoint P of the line segment PP is found.
2. Focus triangle problem
The trigonometric problem formed by a point P on an ellipse or hyperbola and two focal points F and F is usually solved by sine and cosine theorems.
Example: Let P(x, y) be any point of an ellipse += 1, F(-c, 0), and F(c, 0) be the focus. PFF=？ ,? PFF=？ .
(1) verification: eccentricity e =；;
(2) Find the maximum value of |PF|+|PF|.
3. The relationship between straight line and conic curve.
The basic method of the positional relationship between a straight line and a conic curve is to solve the equations, then transform them into a quadratic equation with one variable, and then use the discriminant. Pay special attention to the combination of numbers and shapes.
Example: parabolic equation y = p (x+1) (p >; 0), the intersection of the straight line x+y=t and the x axis is on the right side of the parabola directrix.
(1) proves that there are always two different intersections between a straight line and a parabola.
(2) Let the intersection of a straight line and a parabola be a and b, OA? OB, find the function f(t) of p and the expression of t.
4. Some questions about the maximum value of conic section.
Algebraic methods and geometric methods are often used to solve extreme problems in conic curves. If the conditions and conclusions of the proposition have obvious geometric significance, they can generally be solved by image properties. If the conditions and conclusions of the proposition reflect a clear functional relationship, an objective function (usually quadratic function, trigonometric function and mean inequality) can be established to find the maximum value. In the following question (1), we can first try to get the inequality about a, and then find the value range of a by solving the inequality, namely:? Evaluation domain, inequality? . Or express a as a function of another variable, and find the range of a by finding the range of the function. For (2), we first express the area of △NAB as a function of a variable, and then find its maximum value, that is? Maximum problem, function idea? .
Example: parabola y = 2px (p >; 0), the straight line L passing through M(a, 0) and having a slope of 1 and the parabola are at different points A, B, |AB|? 2p, (1) Find the range of a; (2) If the vertical line of line segment AB intersects with the X axis at point N, find the maximum value of △NAB area.
5. The equation problem of finding the curve
The shape of (1) curve is known, and this kind of problem can generally be solved by undetermined coefficient method.
Example: It is known that the straight line L passes through the origin, the vertex of the parabola C is at the origin, and the focus is on the positive semi-axis of the X axis. If the symmetry points of point A (-1, 0) and point B (0, 8) about L are all on C, find the equation of straight line L and parabola C.
(2) If the curve shape is unknown, find the trajectory equation.
Example: It is known that point Q (2 2,0) and circle C: X+Y = 1 on the rectangular coordinate plane, and the ratio of tangent length from moving point M to circle C to |MQ| is equal to a constant? (? & gt0), find the trajectory equation of moving point m, and explain what curve it is.
6. There are two points about line symmetry.
The symmetry problem of two points on a curve about a straight line can be solved by finding the straight line where the two points are located and finding the intersection point of these two straight lines so that the intersection point is within the shape of a conic curve. Of course, it can also be solved by Vieta theorem combined with discriminant.
Example: Given the equation of ellipse C += 1, try to determine the value range of m, so that for a straight line y=4x+m, there are two different points on ellipse C that are symmetrical about the straight line.
7. The perpendicularity of two line segments
The problem that the two focal radii of a conic curve are perpendicular to each other is in K? K==- 1 to process or use vector coordinate operation to process.