General equation of the circle: expand the standard equation of the circle, shift the terms, and merge the similar terms, and the general equation of the circle can be obtained as X 2+Y 2+DX+EY+F = 0. Compared with the standard equation, in fact, D=-2a, E=-2b and f = a 2+b 2.
The eccentricity of a circle is e=0, and the radius of curvature of any point on the circle is r.
[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, y = (-c-ax)/b, where b is not equal to 0, substitute x 2+y 2+dx+ey+f = 0, that is, it becomes a quadratic equation f(x)=0. Using the symbol of discriminant B 2-4ac, the positional relationship between a circle and a straight line can be determined as follows:
If b 2-4ac > 0, 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.
Equation of straight line and circle
(1) Understand the concept of straight line slope, master the two-point straight line slope formula, master the oblique formula, two-point formula and general formula of straight line equation, and skillfully solve straight line equation according to conditions.
(2) By mastering the condition that two straight lines are parallel and vertical, the angle formed by two straight lines and the distance formula from point to straight line, we can judge the relationship between two straight lines according to the equation of straight lines.
(3) Understand that binary linear inequalities represent plane regions.
(4) Understand the significance of linear programming and apply it simply.
(5) Understand the basic ideas of analytic geometry and coordinate method.
(6) Master the standard equation and general equation of a circle, understand the concept of parametric equation and understand the parametric equation of a circle.
3. When answering questions about straight lines, pay attention to (1). When determining the slope and inclination of a straight line, we should first pay attention to the conditions for the existence of the slope, followed by the range of inclination; (2) When solving problems with the intercept formula of a straight line, attention should be paid to prevent the solution from being lost because of "zero intercept"; (3) When using diagonal lines or diagonal lines to solve problems, we should pay attention to check the absence of inclined planes to prevent the solution from being lost; (4) Flexible use of fixed-point and midpoint coordinate formulas can simplify operations when solving division and symmetry problems; (5) Mastering four basic solutions to symmetry problems; (6) When determining the value or range of related parameters from the positional relationship between two straight lines, we should make full use of basic mathematical thinking methods such as classified discussion, combination of numbers and shapes, and special value test.
Linear equation
1. Tilt angle, slope and direction vector of straight line
(1) inclination angle of straight line
In the plane rectangular coordinate system, for the straight line intersecting with the X axis, if the minimum positive angle when the X axis rotates counterclockwise around the intersection point to coincide with the straight line is recorded as α, then α is called the inclination angle of the straight line.
When the straight line is parallel or coincident with the X axis, we specify that the inclination of the straight line is 0.
It can be seen that the range of linear dip angle is 0 ≤α < 180.
(2) the slope of the straight line
The inclination angle α is not a straight line of 90, and the tangent value of the inclination angle is called the slope of this straight line, which is usually expressed by k, that is, k = tan α (α ≠ 90).
A straight line with an inclination angle of 90 has no slope; All straight lines with an inclination of not 90 have slopes, and their values range from (-∞,+∞).
(3) the direction vector of a straight line
Let F 1(x 1, y 1) and F2(x2, y2) be different points on a straight line, then the vector = (x2-x 1, y2-y 1) is called the direction vector of this straight line. Vector =( 1)
(4) the method of finding the slope of a straight line
① Definition: If the inclination of a straight line is known as α, α ≠ 90, then the slope k=tanα.
② Formula method: a given straight line passes through two points, P 1(x 1, y 1) and P2(x2, y2), x 1≠x2, with a slope of k=.
③ Direction vector method: If a=(m, n) is the direction vector of a straight line, then the slope of the straight line is k = n/m. 。
In the plane rectangular coordinate system, every straight line has an inclination angle, but not every straight line has a slope.
For any two points on the straight line P 1(x 1, y 1) and P2(x2, y2), when x 1=x2, the slope k of the straight line does not exist, and the inclination angle α = 90; When x 1≠x2, the slope of the straight line exists and is a real number. When k≥0, α=arctank, and when k < 0, α=π+arctank.