Mathematics for junior high school Determine the position relationship between a circle and a straig

This problem is very unlucky, if you have studied analytic geometry, it will be good to do it.

The key is to find the coordinates of the center of the circle (it is not difficult to find, you can solve it with junior high school knowledge), then find the radius, and then find the distance from the center of the circle to the straight line that has been translated, and compare the size to determine the position relationship [the straight line in junior high school and the garden are learned].

You can find the analytic formula of the function of the straight line, and then put the items on the side of the equal sign, which was called the equation of the straight line in high school. The formula for finding the distance from the center of the circle to this straight line is: the coefficients of the linear equation are unchanged, substituting the coordinates of the center of the circle into it, finding a value, taking the absolute value, and dividing it by the arithmetic square root of the sum of the squares of the coefficients of x and y, which is this distance.

And then it's better to be older.

Are there any conditions related to straight lines? Arbitrary? If it is an arbitrary straight line, and through a point on the circle, it is either tangent or intersecting, and as for which one, the straight line is arbitrary, it is impossible to judge.

But I don't care what the problem is, since it's in a coordinate system, then you should create one, for example, build a coordinate system with the center of the circle o as the origin, so that the point d is in the third quadrant, and then continue to consider it according to the condition of your straight line.

It's best to send out the original question, you may make a mistake in paraphrasing.

What is point D? Arbitrary?

Methods for judging the positional relationship between a straight line and a circle in junior high school:

1) Algebraic method: determine the position relationship between the straight line ax + by + c = 0 and the circle x2 + y2 + dx + ey + f = 0.

ax+by+c=0

x2+y2+dx+ey+f=0

mx2+nx+p=0 is introduced, using discriminant formula.

Make a judgment. >0 then the line intersects the circle;

0 then the straight line is tangent to the circle;

0 then the line is separated from the circle.

2) Geometric method: the distance from the center of the circle to the straight line is known to be ax+by+c=0 and the circle (x-a)2+(y-b)2=r2.

dd=r, the line is tangent to the circle;

d>r is separated from the circle.

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, and then the straight line is called the secant of the circle.

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.

A straight line and a circle have two common points called "intersections", and this intersecting line is called the secant of the circle. It can be written as ab intersecting with o, d r (d is the distance from the center of the circle to the straight line). Lines and circles have one and only one common point, which is called "tangent".

Write ab tangent to o, d=r.

<> one. Using the distance formula from the point to the line, the distance d from the center of the circle to the line is obtained, and the radius of the circle is r:

1. If d is greater than r, the straight line is separated from the circle;

2. If Bu Xiao D is equal to r, the straight line is tangent to the circle;

3. If d is less than r, the straight line intersects the circle.

Two. A circle is a type of geometric shape. A figure consisting of all points on a plane whose distance to a fixed point is equal to a fixed length is called a circle.

When a line segment rotates in a plane around one of its endpoints, the trajectory of its other end is called a circle. By definition, a circle is usually drawn with a compass.

Question 1: OA=5 3

Question 2: d, the distance to the center of the circle is equal to the radius of the straight line is the tangent of the circle Question 3: c, r

Question 4: Proof: Nexus be, because ab is the diameter.

So be vertical ac

In the RT triangle AEB and the RT triangle BEC.

O and D are the midpoints of the hypotenuse AB and BC, respectively.

So oe=ob

db=de, so again

The relationship between a straight line and a circle: apart, tangent, intersecting.

Separation: d>r

Tangent: d=r

Intersection: d

Suppose the center of the circle is (x1,y1).

1) The circle p and the x-axis are tangent, indicating y1=2, and p(3 2,2) is obtained;

2) The circle p and y axis are tangent, indicating x1=2;p(2,3);

3) Since the radius of the circle is 2, if the circle is tangent to both the x-axis and the y-axis, then x1=y1, giving x1=y1=1<2, so it can't.

In this figure, the radius of the inscribed circle = 1 3 equilateral triangles are high.

And equilateral triangle height = 2 * cos30° = 3

So the radius is 3 3 long (just read it wrong, sorry).

The side length is 2, then the height is the root number 3, and OA is equal to 2 times OD, so OD is equal to one-third of the root number 3, and the radius is one-third of the root number 3

According to the triangle odc is a right triangle, the angle ocd is 30°, so 0d is equal to cd root number 3, cd = 1, so od = root number 3 3

In the Cartesian coordinate system xoy, when the coordinate point falls on the x-axis, y=0 (x belongs to the real number, and the value of x can get the coordinate point)When the coordinate point falls on the y-axis, x=0 (y belongs to the real number, and the value of y can get the coordinate point) The coordinates on the coordinate axis can be determined through the above functional relationship, as long as the value of x or y is taken, it is OK

From the question: a, b, and c are the tangent points of the circle o.

pa=pc , qb=qc

pq=pc+cq=ap+bq

In RT PHQ, PQ 2=PH 2+QH 2 (AP+BQ) 2=AB 2+(BQ-AP) 2 AB 2=4AP x BQ

PA and PC are tangents of the circle O.

apo=∠cpo=∠apc/2

In the same way: bqo = cqo = bqc 2

again a= b=90°

pa‖qb ∠apc+∠bqc=180°

cpo+∠cqo=(∠apc+∠bqc)/2=90°∴∠poq=90°

poa+∠qob=90°

and poa+ opa=90°

opa=∠qob

again a= b=90°

rt△pao∽rt△obq

ao/bq=ap/bo

ao*bo=ap*bq

ao=bo=ab/2

ab/2*ab/2=ap*bq

ab^2=4ap x bq