Position relationship between circle and point: Take point P and circle O as an example (let P be a point, then PO is the distance from the point to the center of the circle), where P is outside ⊙O, and PO > R；; P on ⊙O，po = r； P is less than ⊙O, and 0 ≤ PO < R. There are three kinds of positional relationships between straight lines and circles: there is no separated common point; There are two common points intersecting, and this straight line is called the secant of the circle; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point. Take straight line AB and circle O as examples (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle): AB is separated from ⊙O, and po > r；; AB is tangent to ⊙O, po = r；; AB intersects with ⊙O, and 0 ≤ PO < R. There are five positional relationships between two circles: if there is no common point, one circle is called external separation and internal inclusion outside the other circle; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; Including P 0). Compared with the standard equation, in fact, D=-2a, E=-2b and f = a 2+b 2-r 2. The center coordinates of the circle are (-D/2, -E/2) and the radius r = 0.5 √ d 2+e 2-4f. Parametric equation of circle: The parametric equation of a circle with point O(a, b) as the center and R as the radius is x=a+r*cosθ, y=b+r*sinθ, where θ is the parameter. Endpoint formula of a circle: two points A (A 1, B 1) and B (A2) are known, and the tangent equation of a point M(a0, b0) passing through a circle X 2+Y 2 = R 2 is A0 * X+B0 * Y = R 2. A point M(a0, B0) outside the circle (X 2+Y 2 = R 2) leads to two tangents of the circle, which are A and B.
Judgment of the positional relationship between circle and straight line
In the plane, the general method to judge the positional relationship between the straight line Ax+By+C=0 and the circle X 2+Y 2+DX+EY+F = 0 is 1. From Ax+By+C=0, substitute Y = (-C-AX)/B, where b is not equal to 0. Using the sign of discriminant B 2-4ac, we can determine the positional relationship between circle and straight line as follows: If B2-4ac >；; 0, a circle and a straight line have two intersections, that is, a circle and a straight line intersect. If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line. If b 2-4ac
"Circle" in Mathematics
[Definition of circle]
Geometrically, a figure composed of all points whose distance from a plane to a fixed point is equal to a fixed length is called a circle. A fixed point is called the center of the circle and a fixed length is called the radius.
Trajectory theory: the trajectory of a moving point on a plane with a certain length as the center is called a circle.
Set theory: The set of points whose distance to a fixed point is equal to a fixed length is called a circle.
Correlation quantity of circle
Pi: The ratio of the circumference of a circle to its diameter and length is called pi, and its value is 3.14159265358979323846 …, which is usually expressed by π. In calculation, 3. 14 16 is often taken as its approximate value.
Arc chord: the part between any two points on the circle is called arc, or simply arc. An arc larger than a semicircle is called an upper arc, and an arc smaller than a semicircle is called a lower arc. A line segment connecting any two points on a circle is called a chord. The chord passing through the center of the circle is called the diameter.
Central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle.
Inner and outer center: the circle passing through the three vertices of the triangle is called the circumscribed circle of the triangle, and its center is called the outer center of the triangle. A circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, and its center is called the heart.
Sector: On a circle, the figure enclosed by two radii and an arc is called a sector. The development diagram of the cone is a sector. The radius of this sector becomes the generatrix of the cone.
Letter representation of correlation between circles
Circle-⊙ radius -R arc-⌒ diameter-D.
Sector arc length/conical generatrix -l circumference -c area -s
[positional relationship between circle and other figures]
Position relationship between circle and point: Take point P and circle O as an example (let P be a point, then PO is the distance from the point to the center of the circle), where P is outside ⊙O, and PO > R；; P on ⊙O，po = r； P is within ⊙O, and po < r.
There are three positional relationships between a straight line and a circle: there is no separated common point; There are two things in common; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point. Take straight line AB and circle O as examples (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle): AB is separated from ⊙O, and po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
There are five positional relationships between two circles: if there is nothing in common, one circle is called external separation and internal inclusion outside the other; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains P 0, then the circle and the straight line have two intersections, that is, the circle and the straight line intersect.
If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line.
If b 2-4ac
2. If B=0 indicates that the straight line is Ax+C=0, that is, X =-C/A, parallel to the Y axis (or perpendicular to the X axis), change X 2+Y 2+DX+EY+F = 0 to (X-A) 2+(Y-B) 2 = R, and let Y =
When x =-c/a X2, the straight line is separated from the circle.
When x 1
When x =-c/a = x 1 or x =-c/a = x2, the straight line is tangent to the circle.