1. The tangent PT at point P bisects the outer angle of △PF 1F2 at point P. 。

2. If PT bisects the external angle of △PF 1F2 at point P, then the trajectory of the projected H point with the focus on the straight line PT is a circle with the diameter of the long axis, and the two endpoints of the long axis are removed.

3. The circle with the diameter of focus chord PQ must be separated from the corresponding directrix.

4. The circle with the focal radius PF 1 must be inscribed with the circle with the diameter of the long axis.

5. If it is on an ellipse, the tangent equation of the crossed ellipse is.

6. If outside the ellipse, the tangent points of two tangents passing through Po are P 1 and P2, and the linear equation of tangent chord P 1P2 is.

7. the left and right focal points of an ellipse (A > B > 0) are F 1 and F 2 respectively, and the point p is any point on the ellipse, then the focal angle area of the ellipse is.

8. the formula of focal radius of ellipse (a > b > 0):

, ( , ).

9. Let the focus F of the ellipse be the points P and Q where the straight line intersects the ellipse, A be the vertex on the long axis of the ellipse, and the connecting lines AP and AQ intersect the ellipse directrix corresponding to the focus F at points M and N, then MF⊥NF.

10. The straight line passing through the focus f of the ellipse intersects the ellipse at two points P and Q, where A 1 and A2 are the vertices on the long axis of the ellipse, A 1P and A2Q intersect at point M, A2P and A 1Q intersect at point N, and then MF⊥NF.

1 1.AB is the chord of an ellipse that is not parallel to the axis of symmetry, and m is the midpoint of AB, then,

Namely.

12. If it is in an ellipse, the equation that the midpoint chord is bisected by Po is.

13. If it is in an ellipse, the locus equation of the midpoint of the chord passing through Po is.

hyperbola

1. The tangent PT at point P bisects the internal angle of △PF 1F2 at point P. 。

2. If PT bisects the internal angle of △PF 1F2 at point P, then the trajectory of the projected H point with the focus on the straight line PT is a circle with the diameter of the long axis, and the two endpoints of the long axis are removed.

3. The circle with the diameter of focus chord PQ must intersect with the corresponding directrix.

4. The circle whose diameter is the focal radius PF 1 must be tangent to the circle whose diameter is the real axis. Circumscribed: p in the left branch)

5. if it is on a hyperbola (a > 0, b > 0), the tangent equation of hyperbola is.

6. If it is outside the hyperbola (A > 0, B > 0), the tangent points of two tangents passing through Po as hyperbola are P 1 and P2, and the linear equation of tangent chord P 1P2 is.

7. The left and right focal points of hyperbola (A > 0, B > 0) are F 1 and F 2 respectively, and point P is any point on hyperbola, then the focal angle area of hyperbola is.

8. formula of focal radius of hyperbola (a > 0, b > 0): (,

When in the right branch.

When in the left branch,

9. Let the focus f of hyperbola be point P and point Q where the straight line intersects with hyperbola, point A be a vertex on the long axis of hyperbola, and the hyperbola directrix connecting AP and AQ intersect at point M and point N respectively, then MF⊥NF.

10. The straight line passing through the hyperbola focus f intersects the hyperbola at two points p and q, A 1 and A2 are the vertices on the real axis of the hyperbola, A 1P and A2Q intersect at point m, A2P and A 1Q intersect at point n, and then MF⊥NF.

1 1.AB is the chord of hyperbola (a > 0, b > 0) that is not parallel to the symmetry axis, and m is the midpoint of AB, that is.

12. if it is in hyperbola (A > 0, B > 0), the equation that the midPoint chord is bisected by po is.

13. if it is in hyperbola (A > 0, B > 0), the locus equation of the chord midPoint passing through po is.

Dual properties of ellipse and hyperbola-(a classical conclusion that can be deduced)

Elliptic circle

The two vertices (A > B > O) of the ellipse 1. are 0, and the straight line parallel to the Y axis intersects the ellipse at P 1. When P2 is 0, the trajectory equation of A 1P 1 intersects with A2P2.

2. Make two straight lines (A > 0, B > 0) with complementary inclination angles at any point on the ellipse, and intersect the ellipse at points B and C, then the BC line is directional and (constant).

3. if p is any point on the ellipse (a > b > 0) different from the endpoint of the long axis, and F 1 and F 2 are the focus,,, then.

4. Let the two foci of the ellipse (a > b > 0) be F 1, and F2, p (different from the endpoint of the long axis) be any point on the ellipse. In △PF 1F2, notice,,, and then you have it.

5. if the left and right focus of an ellipse (a > b > 0) are F 1 and F2 respectively, and the left directrix is l, then if 0 < e ≤, we can find a point p on the ellipse, so PF 1 is the middle term of the ratio of p to the corresponding directrix distances d and PF2.

6. if p is any point of an ellipse (a > b > 0), F 1, F2 is two focal points, and a is a fixed point in the ellipse, then the equal sign holds if and only if it is a three-point * * * line.

7. An ellipse and a straight line have a common point if and only if.

8. Known ellipse (A > B > 0), O is the origin of coordinates, P and Q are two moving points on the ellipse, and (1); (2) The maximum value | op | 2+| OQ | 2 is; The minimum value of (3) is.

9. if the right focus f of an ellipse (A > B > 0) intersects a straight line, the right branch of the ellipse is at m and n points, and the perpendicular line of the chord MN intersects the x axis at p, then.

10. given an ellipse (a > b > 0), a and b are two points on the ellipse, and the perpendicular of the line segment AB intersects the x axis at this point, then.

1 1. Let point P be any point on the ellipse (a > b > 0) different from the endpoint of the long axis, and F 1 and F2 are its focal points, then (1).

12. let a and b be the two ends of the long axis of the ellipse (A > B > 0), p be a point on the ellipse, and,,, c and e be the heart rate of the ellipse's half focal length, respectively, then there is (1). (2) .(3).

13. it is known that the right directrix of the ellipse (a > b > 0) intersects with the x axis at a point, and the straight line passing through the right focus of the ellipse intersects with the ellipse at points a and b, which are on the right directrix, and the axis, then the straight line AC passes through the midpoint of the line segment EF.

14. If the endpoint passing through the radius of the focal point of the ellipse is the tangent of the ellipse and intersects the circle with the diameter of the long axis, the connecting line between the corresponding intersection point and the corresponding focal point must be perpendicular to the tangent.

15. If the tangent of the ellipse intersects the corresponding directrix at a point through the endpoint of the focal radius of the ellipse, the connecting line between the point and the focal radius must be perpendicular to the focal radius.

16. In an elliptical focal triangle, the ratio of the distance from the inner point to a focal point to the focal radius with the focal point as the endpoint is a constant e (eccentricity).

(Note: In an elliptical focus triangle, the intersection of the bisector of the inner angle and the outer angle of the defocused vertex with the long axis is called the inner point and the outer point respectively. )

17. In the ellipse focus triangle, the inner heart divides the line between the inner point and the defocused vertex into a constant ratio e. 。

18. In an elliptical focal triangle, the half focal length must be the average of the ratio of the inner and outer points to the center of the ellipse.

Dual properties of ellipse and hyperbola-(a classical conclusion that can be deduced)

hyperbola

The two vertices (A > 0, B > 0) of the hyperbola 1. are, and when the hyperbola intersects a straight line parallel to the y axis, the trajectory equation of the intersection of A 1P 1 and A2P2 is.

2. If any point on the hyperbola (A > 0, B > 0) intersects with two straight lines with complementary inclination angles at B and C, the straight line BC is directional and (constant).

3. If p is any point on the right (or left) branch of hyperbola (A > 0, B > 0) except the vertex, F 1, F 2 is the focus,,, then (or).

4. Let the two focal points of hyperbola (A > 0, B > 0) be F 1, and F2, P (different from the long axis endpoint) be any point on hyperbola. In △PF 1F2, notice,,, and then you have it.

5. If the left and right foci of hyperbola (A > 0, B > 0) are F 1 and F2, respectively, and the left directrix is L, then when 1 < E ≤, we can find a point p on hyperbola, so PF 1 is the median of the ratio of p to the corresponding directrix distances d and PF2.

6. if p is any point on the hyperbola (A > 0, B > 0), F 1, F2 is two focal points, and a is a fixed point on the hyperbola, then the equal sign holds if and only if the three-point * * * line is on the same side of the y axis.

7. The hyperbola (a > 0, b > 0) and the straight line have a common point if and only if.

8. given hyperbola (b > a > 0), o is the origin of coordinates, p and q are two moving points on hyperbola, and.

( 1) ; (2) The minimum value of 2)| op | 2+| OQ | 2 is; The minimum value of (3) is.

9. the right focus f of hyperbola (A > 0, B > 0) is a straight line intersecting with the right branch of hyperbola at point m and point n, and the perpendicular line of chord MN intersects with axis x at point p, then.

10. given a hyperbola (A > 0, B > 0), a and b are two points on the hyperbola, and the perpendicular of the line segment AB intersects the x axis at this point, then or.

1 1. Let point P be any point on the hyperbola (a > 0, b > 0) that is different from the endpoint of the real axis, and F 1 and F2 are its focus, then (1). (2).

12. Let A and B be the two ends of the long axis of hyperbola (A > 0, B > 0), P is a point on hyperbola, and,,, C and E are the half-focal heart rate of hyperbola, so there is (1).

(2) .(3) .

13. it is known that the right directrix of hyperbola (a > 0, b > 0) intersects with the x axis at a point, and the straight line passing through the right focus of hyperbola intersects with hyperbola at points a and b, which are on the right directrix and axis, then the straight line AC crosses the midpoint of line segment EF.

14. If the endpoint passing through the focal radius of hyperbola is the tangent of hyperbola and intersects with a circle whose long axis is diameter, the connecting line between the corresponding intersection point and the corresponding focal point must be perpendicular to the tangent.

15. If passing through the endpoint of the focal radius of hyperbola makes the tangent of hyperbola intersect the corresponding directrix at a point, then the connecting line between this point and the focal radius must be perpendicular to each other.

16. In a hyperbolic focal triangle, the ratio of the distance from the outer point to a focal point to the focal radius with the focal point as the endpoint is a constant e (eccentricity).

(Note: In the hyperbolic focus triangle, the intersection of the bisector of the inner and outer angles of the defocused vertex and the long axis is called the inner and outer points respectively).

17. In the hyperbolic focus triangle, the auxiliary center opposite to the focus divides the line between the outer point and the defocused vertex into a constant ratio e. 。

In the hyperbola focus triangle, the half focal length must be the proportional average of the inner and outer points to the hyperbola center.