Mathematical line and circle position problems, the position relationship between lines and circles

p1(1 2,-1) is substituted for a(n+1)=((6an)+5)) ((4an)+6),b(n+1)=-(2bn) (2an+3)(n n); Gotta :

a2=1;b2=1/2;i.e.: p2(1,1 2).

p2(1,1 2) substitution:

a(n+1)=((6an)+5))/((4an)+6),b(n+1)=-(2bn)/(2an+3)(n∈n）;Gotta :

a3=11/10;b2=-1/5;i.e.: p3(11 10,-1 5);

Let the circular equation of p1p2p3 be x 2 + y 2 + dx + eyy + f = 0;

Substituting the coordinates of three points, solving the system of equations obtains: d=0; e=0;f=-5/4;

That is, the equation for the circle m is x 2 + y 2 = 5 4;

2) The position of pn is on the circle m;

Proof: (Mathematical Induction).

When n=4; Substitute p(11 10,-1 5) into a(n+1)=((6an)+5)) ((4an)+6), b(n+1)=-(2bn) (2an+3)(n n); Gotta :

a4=29/26;b4=1/13;

a4^2+b4^2=5/4;The proposition is true;

Let n=k; The proposition is true; i.e. ak 2 + bk 2 = 5 4; bk^2=5/4-ak^2;

when n=k+1;

a(n+1)=((6an)+5))/((4an)+6),b(n+1)=-(2bn)/(2an+3)(n∈n）;Gotta :

a(k+1)^2=(6ak+5)^2/4(2ak+3)^2=(36ak^2+60ak+25)/4(2ak+3)^2;

b(k+1)^2=4bk^2/(2ak+3)^2=4*(5/4-ak^2)/(2ak+3)^2;

a(k+1)^2+b(k+1)^2=(20ak^2+60ak+45)/4(2ak+3)^2=5(2ak+3)^2/4(2ak+3)^2

5/4;The proposition is proven; pn on the circle m;

The conclusion is that it cannot be at a 45-degree angle;

Counter-proof: Let the tilt angle of APN be 45 degrees; Then the linear equation for APN is y=x+ 5;

As evidenced above, pn is on the circle m; Substitute y=x+5 into x2+y2=5 4; Gotta :

x^2+(x+√5)^2=5/4;

2x^2+2√5x+15/4=0;

Discriminant =(2 5) 2-4*2*(15 4)=20-30=-10<0; No solution;

Therefore, there is no pn point to satisfy the apn inclination angle of 45 degrees;

Tangent proofs can also be used; The minimum inclination of the APN is 60 degrees; )

This is the most basic thing in mathematics, so let's find the expressions for an and bn separately, and then bring them in.

There are three positional relationships between lines and circles, as follows:

1. Intersection: When a straight line and a circle have two common points, it is called the intersection of a straight line and a circle, then the straight line is called the secant of the circle, and the common point is called the intersection point.

2. Tangent: When a straight line and a circle have a single common point, it is called a tangent between a straight line and a circle, and then a straight line is called a tangent of a circle.

3. Separation: When there is no common point between the straight line and the circle, it is called the separation of the straight line and the circle.

If the radius of the circle o is r and the distance from the center of the circle o to the line l is d, then:

When the line l intersects the circle o, d r.

When the line l is tangent to the circle o, d=r.

The distance between the line l and the circle o is, d>r.

The 4 question types of the Straight Line and Circle Common Exam:

Type 1: Determination of the position relationship between a line and a circle.

Type 2: The nature of the tangent of the circle.

If there are tangents in the circle, the radius of the tangent point is often connected to construct a right-angled triangle, and then the degree of the angle is found in the right-angled triangle, or the length of the line segment is found using the Pythagorean theorem.

Type 3: Determination of tangents.

When proving that a straight line is a tangent of a circle, if it is known that the straight line has a common point with the circle, the radius of the point can be made to prove that the straight line is perpendicular to the radius, that is, "as a radius, it is proved perpendicular"; If it is not certain that a straight line has a common point with a known circle, then the perpendicular segment of the straight line is made through the center of the circle, proving that the distance from it to the center of the circle is equal to the radius, that is, "as perpendicular, the radius is certified".

Type 4: The inscribed circle and tangent length theorem of triangles.

A closed curve formed by rotating around a point at a distance of a certain length in a plane is called a circle. The positional relationship between a straight line and a circle is distancing, intersecting, and tangent. There are two ways to determine this:

One is judged by the number of common points between the straight line and the circle: the straight line and the circle have no common points, which is called separation; A straight line and a circle have two common points called intersections, and this line is called the secant of the circle; Lines and circles have one and only one common point, which is called tangent. This straight line is called the tangent of the circle, and this common point is called the tangent point, and the line connecting the center of the circle with the tangent point is perpendicular to the tangent line.

The second is to judge from the relationship between the distance from the center of the circle to the straight line and the radius: let the radius of the circle be r, and the distance from the center of the circle to the straight line is d, then the conclusion is:

Distancing: d r; Tangent: d=r; Intersect: d r.

One is the equation of a straight line and a circle, which is judged by the number of solutions (two solutions, intersecting, one solution, tangent, no solution, distancing). The other is to obtain the coordinates of the center of the circle, compare the distance from the center of the circle to the straight line with the radius of the circle, greater than the radius means that it is far away, equal to the radius of the tangent, less than the radius means intersecting.

The relationship between the position of a straight line and a circle is a part of the analytic geometry content of high school mathematics, and the exam mainly involves the equations of straight lines, the equations of circles, and the position relationship between lines and circles. It requires a certain amount of computing ability, imagination ability, logical reasoning ability, and drawing ability. Medium difficulty.

The position of the line and the circle is as follows: d=|am+bn+c|/√a^2+b^2)。

1. If there is no common point between the straight line and the circle, then the position relationship between the straight line and the circle is called separation.

2. If there is only one common point between the straight line and the circle, then the positional relationship between the straight line and the circle is called tangent, the straight line is called the tangent of the circle, and the common point is called the tangent point.

3. If there are two common points between the straight line and the circle, then the position relationship between the straight line and the circle is called intersection, and the straight line is called the secant of the circle.

1. The area of the semicircle: s semicircle = (r 2) 2. (r is the radius).

2. The area of the ring: S big circle - S small circle = (r 2-r 2) (r is the radius of the big circle of Blind Changna, and r is the radius of the small circle).

3. The circumference of the circle: c=2 r or c=d. (d is the diameter, r is the radius).

4. The circumference of the semicircle: d+(d) 2 or d+ r. (d is the diameter, r is the radius).

5. The length of the fan arc l = the central angle (arc speed dislike system) r = n r 180 (for the grinding of the central angle) (r is the fan radius).

6. The sector area s=n r 360=lr 2 (l is the arc length of the fan).

1. Tangent. When a straight line and a circle have a single common point, it is called tangent between a straight line and a circle.

2. Separation. When a straight line and a circle have no common point, it is said that the line and circle are separated.

3. Intersection. When a line and a circle have two common points, it is called the intersection of the line and the circle.

Here's how to determine the position relationship between a line and a circle:

1. Judge whether there is a public point.

The straight line is separated from the circle, and there is no common point; The straight line is tangent to the circle and has only one common point; A straight line intersects a circle and has two common points. In a plane, a closed curve formed by a moving point centered on a certain point and rotated around a certain length is called a circle. A straight line is made up of an infinite number of points.

The straight line is the component of the surface, and the composition is followed by the combustion of the ruler. There are no endpoints, and the length is inmeasurable and extends indefinitely to both ends. A straight line is an axisymmetric figure.

It has an infinite number of axes of symmetry, one of which is itself, and all the straight lines (with an infinite number of axes) perpendicular to it. If there is only one straight line at two points that do not coincide on the plane, that is, two points that do not coincide determine a straight line. On a spherical surface, crossing two points can make an infinite number of similar straight lines.

2. Straight line method.

If the straight line is always past the fixed point, it can be judged by judging the position relationship between the point and the circle, but it has certain limitations, and it must be a straight line system that is over the fixed point.

3. Algebraic method.

Simultaneous linear equations and circular equations, solve the system of equations, and if the system of equations has no solution, then the straight line is separated from the circle. If the system of equations has 1 set of solutions, then the trapped core of the line is tangent to the circle, and if the system of equations has 2 sets of solutions, the line intersects the circle.

4. Geometric method.

Find the distance d from the center of the circle to the straight line, the radius is 》r, then the straight line is separated from the circle, d=r, then the straight line is tangent to the circle, and d is <>

1. The positional relationship between a straight line and a circle.

There are three kinds of positional relationships between straight lines and circles, which are intersecting, distancing, and tangent. In general, when there is no common point between a straight line and a circle, it is said that the straight line is separated from the circle; When a straight line has a unique common point with a circle, it is called a tangent between a straight line and a circle; When a line and a circle have two points in common, it is called a line intersecting a circle.

2. Knowledge Expansion - Theorem.

If the radius of o is r and the distance from the center of the circle o to the line l is d (r 0 and d 0), then:

d r 0 straight line l is separated from o;

d=r 0, the line l is tangent to o; When a straight line is tangent to a circle, the straight line is called the tangent of the circle, and this only common point is called the tangent point. The line connecting the center of the circle to the tangent point is perpendicular to the tangent.

0 d r line l intersects o . (d=0, the line just passes through the center of the circle) When a straight line intersects a circle, the straight line is called the secant of the circle.

4 types Draw a circle with C as the center of the circle, and the arc is tangent to the AB edge (the radius of the sector edge is on AC, BC).

Draw a circle with A as the center of the circle, and the arc is tangent to the BC edge (the radius of the sector edge is on AC, AB) A point tangent to BC, AB can be found with a point on the AC edge as the center of the circle (the radius of the sector edge is on AC).

Draw a circle with the midpoint of AB as the center of the circle, and the arc is tangent to AC and BC (the radius of the sector edge is on AB).

Detailed enough...

Solution: When the chord ab is bisected by the point p, obviously.

AB is bisected perpendicularly by the diameter op.

The analytic formula for finding the straight line op is: y

2x Therefore, the analytic formula for the straight line ab is: y-2

2(x+1)

Asking for the length of the string is trivial, I don't think it needs to be said!

1. Tangent.

When a straight line and a circle have a single common point, it is called a tangent between a straight line and a circle.

2. Separation. When a straight line and a circle have no common point, it is said that the line and circle are separated.

3. Intersection. When a line and a circle have two common points, it is called the intersection of the line and the circle.

There are three kinds of positional relationships between straight lines and circles: intersecting, tangent, and distancing.

Intersection, Chinese vocabulary. It is interpreted as two straight lines that intersect each other and intersect at one point. Make friends; Be friends.

If a straight line and a curve intersect at two points, and the two points are infinitely close and tend to coincide, the straight line is the tangent of the curve at that point. In junior high school mathematics, if a straight line is perpendicular to the radius of a circle and passes the outer end of the radius of a circle, the line is said to be tangent to the circle.

Tangency is a positional relationship between a circle on a plane and another geometric shape.

To be separated is to be separated from one another.

How to judge the position relationship between a straight line and a circle:

1. Algebraic method:

Simultaneous linear equation and circle equation, solve the system of equations, if the system of equations has no solution, then the straight line is separated from the circle, if the system of equations has 1 set of solutions, then the straight line is tangent to the circle, and if the system of equations has 2 sets of solutions, then the line intersects the circle.

2. Geometric method:

Find the distance d from the center of the circle to the straight line, the radius is 》r, then the line is separated from the circle, d=r, then the line is tangent to the circle, d

Intersect and Beyond the Center of the Circle: The line intersects the circle and the center of the circle is on the line.

Intersect but not center of the circle": The line intersects the circle, and the center of the circle is not on the line.