1. There are two common methods to solve the range problem of conic curves:?

(1) To find a reasonable inequality, it is common that the midpoint of the chord △ > 0 is inside the curve; ?

(2) The required quantity can be expressed as a function of another variable, and the range of the function can be found. ?

2. Problems such as maximum value, fixed value and crossing point of conic curve.

(1) From a geometric point of view, there are three positional relationships between straight lines and conic curves: separation, tangency and intersection. Separation means that straight lines and conic curves have no common points, tangency means that straight lines and conic curves have only common points, and intersection means that straight lines and conic curves have two different common points. It is important to note that straight lines, hyperbolas and parabolas only have something in common. When a straight line is parallel (coincident) with the parabola's symmetry axis, it has a unique common point with the parabola, but at this time the straight line intersects with the parabola, so when it has a unique common point with the hyperbola and parabola, it may be tangent or intersect. When a straight line intersects these two curves, there may be two intersections or one intersection, so it is not necessary to judge the positional relationship between the straight line and the curve by using common points.

(2) From the algebraic point of view, the positional relationship can be determined according to the number of solutions of the equation group composed of straight line equation and conic curve equation. Let the equation of straight line L be simultaneous with the conic equation to get AX2+BX+C = 0.

(1) If a=0, when the quadratic curve is a hyperbola, the straight line L is parallel or coincident with the asymptote of the hyperbola; When the conic curve is a parabola, the straight line L is parallel or coincident with the symmetry axis of the parabola.

② If

When δ > 0, the straight line and conic curve intersect at two different points.

When δ = 0, both the straight line and the conic are tangent to a point.

When δ

Chord length formula of intersection of straight line and conic curve;

If the straight line L and the quadratic curve F(x, y)=0 intersect at point A and point B, the chord length AB can be found in the following two ways:

(1) Method of finding the intersection point: Combine the equation of a straight line with the equation of a quadratic curve to get the coordinates of point A and point B, and then use the distance formula between the two points to get the length of the chord AB. Generally speaking, this method is more troublesome.

(2) Vieta Theorem Law:

(1) From a geometric point of view, there are three positional relationships between straight lines and conic curves: separation, tangency and intersection. Separation means that straight lines and conic curves have no common points, tangency means that straight lines and conic curves have only common points, and intersection means that straight lines and conic curves have two different common points. It is important to note that straight lines, hyperbolas and parabolas only have something in common. When a straight line is parallel (coincident) with the parabola's symmetry axis, it has a unique common point with the parabola, but at this time the straight line intersects with the parabola, so when it has a unique common point with the hyperbola and parabola, it may be tangent or intersect. When a straight line intersects these two curves, there may be two intersections or one intersection, so it is not necessary to judge the positional relationship between the straight line and the curve by using common points.

(2) From the algebraic point of view, the positional relationship can be determined according to the number of solutions of the equation group composed of straight line equation and conic curve equation. Let the equation of straight line L be simultaneous with the conic equation to get AX2+BX+C = 0.

(1) If a=0, when the quadratic curve is a hyperbola, the straight line L is parallel or coincident with the asymptote of the hyperbola; When the conic curve is a parabola, the straight line L is parallel or coincident with the symmetry axis of the parabola.

② If

When δ > 0, the straight line and conic curve intersect at two different points.

When δ = 0, both the straight line and the conic are tangent to a point.

When δ

Chord length formula of intersection of straight line and conic curve;

If the straight line L and the quadratic curve F(x, y)=0 intersect at point A and point B, the chord length AB can be found in the following two ways:

(1) Method of finding the intersection point: Combine the equation of a straight line with the equation of a quadratic curve to get the coordinates of point A and point B, and then use the distance formula between the two points to get the length of the chord AB. Generally speaking, this method is more troublesome.

(2) Vieta Theorem Law:

If the coordinates of the intersection point are not found, it can be solved by Vieta theorem. If the equation of straight line L is expressed by y=kx+m or x=n 。