Circle O1 the radius of circle O2 is R.R, the distance between centers of circle is D, and the two circles are outward. Point P moves on circle O1, and point Q moves on circle O2. The maximum of PQ is asked The radius of circle O1 and circle O2 are R.R, D and D respectively. Point P moves on circle O1 and point Q moves on circle O2. What are the maximum and minimum values of PQ and why? I already know the answer. = = if you want to score, please answer it carefully!

(2002, Guangzhou) if the radii of ⊙ O1 and ⊙ O2 are 1 and 3 respectively, and ⊙ O1 and ⊙ O2 are circumscribed, then the circle with radius 4 on the plane and tangent to ⊙ O1 and ⊙ O2 has () A. 2 B. Three C. Four D. Five

The radii of ⊙ O1 and ⊙ O2 are 1 and 3 respectively, and the radius is 4,
1+3=4，
There are 5 circles tangent to ⊙ O1 and ⊙ O2;
There are two circles which are tangent to the two circles, one of which is circumscribed and the other is inscribed; the other is that the two circles are tangent to the two circles, each of which is circumscribed
Therefore, D

Given that the circle O1: (x + 3) ^ 2 + y ^ 2 = 1 and O2: (x-3) ^ 2 + y ^ 2 = 9, the moving circle is circumscribed with two circles at the same time, and the trajectory equation of the center of the moving circle is obtained

Let the center of the moving circle be m and the radius of the moving circle R
Then | o1m | = R + 1, | o2m | = R + 3
|O2M|-|O1M|=2
So the trajectory of M is one of the hyperbolas focusing on O1, O2,
It's far from O2, so it's the left branch
c=3,a=1
b²=9-1=8
Therefore, the trajectory equation of the center of a moving circle is x? Y? 2 / 8 = 1 (x ≤ - 1)

Given that the moving circle m and the circle O1: x ^ 2 + (Y-1) ^ 2 = 1 and the circle O2: x ^ 2 + (y + 1) ^ 2 = 4 are circumscribed, find the locus square of the center m of the moving circle

First you draw the scene on the sketch, and I'll tell you more quickly
It can be seen from the graph that | mo2 | - | mo1 | = 1. For a fixed value, we can associate that the trajectory is actually a hyperbola
Let m: x ^ 2 / A ^ 2-y ^ 2 / b ^ 2 = 1. Then 2C = 2, 2A = 1
∴b^2=3/4,
Therefore, M: x ^ 2 / (1 / 4) - y ^ 2 / (3 / 4) = 1

If the moving circle P is inscribed with the circle O1: x2 + y2 = 1 and the circle O2: x2 + y2-8x + 7 = 0, then the locus of the center of the moving circle P is______ .

If the radius of the moving circle is r, then | po1 | + 1 = R, | PO2 | + 3 = R,
∴|PO1|-|PO2|=2，
The locus of the center of the moving circle P is the right branch of the hyperbola with O1 and O2 as the focus and the center at (1.5, 0),
So the answer is: O 1, O 2 as the focus, the center in (1.5, 0) of the right branch of the hyperbola

Given that the length of a chord is equal to the radius r, find: (1) The length of the inferior arc to which the string is directed; (2) The area of the bow formed by this chord and minor arc

(1) As shown in the figure, if the chord ab of ⊙ o with radius is r, then ⊙ OAB is an equilateral triangle, so ∠ AOB = π 3, then the minor arc of chord AB is π 3R (3 points) (2) because s △ AOB = 12 · OA · obsin ∠ AOB = 34r2, s sector AOB = 12 | α | R2 = 12 × π 3 × R2 = π 6r2, so s bow = s fan

A chord of a circle is equal to the radius, and the angle of the center of the circle that this string is opposite is______ Degree

If the radius is r, then the chord length is r,
The chord can form an equilateral triangle with an inner angle of 60 °,
Therefore, the angle of the center of the circle to which the string is opposite is 60 degrees
So the answer is 60

In rectangular coordinate system, point a (6,0), point B (0,8), point C (- 4,0), point m starts from C and moves towards point a at the speed of 2 units per second along the direction of Ca; point n starts from point a and moves at the speed of 5 units per second along AB direction, the intersection point of Mn and Y axis is p, and points m and N start to move at the same time. When point m reaches point a, the motion stops. In the process of motion, the time of motion is T seconds (1) When t is, Mn ⊥ ab (2) Whether the value of MP / PN will change during the movement of point m from point C to point O (excluding point o), if not, try to find out the constant value, and if it will change, explain the reason; (3) Is it possible for △ BPN to be an isosceles triangle during the whole movement? If so, try to find out the corresponding value of T. if not, explain the reason

(1) The slope of the straight line AB is: k = (0-8) / (6-0) = - 4 / 3. If Mn is perpendicular to AB, the slope of Mn is: - 1 / k = 3 / 4. According to the meaning of the question, the coordinates of M and N are m (- 4 + 2T, 0) and n (6-3t, 4T), so (4t-0) / [(6-3t) - (- 4 + 2t)] = 3 / 4

Given the point Q (2,0) and the circle C: X ∧ 2 + y ∧ 2 = 1 on the plane of rectangular coordinate system, the ratio of tangent length from moving point m to circle C and │ MQ │ is equal to the constant λ, λ > 0 Try to find the trajectory equation of moving point m and explain what curve it represents

As shown in the figure, if Mn is tangent to N, then the set composed of moving point m is p = {m | Mn | = | MQ}, with a constant > 0
∵ radius of circle | on | = 1
∴||MN|2 = |MO|2－|ON|2 = |MO|2－1
If the coordinates of point m are (x, y), then=
(2-1) (x2 + Y2) - 42x + (1 + 42) = 0
When = 1, the equation is x =, representing a straight line
When ≠ 1, the equation is (x -) 2 + Y2=
It represents a circle with a center of (, 0) and a radius of

As shown in the figure, in the rectangular coordinate system, point m is in the first quadrant, Mn ⊥ X axis is at point n, Mn = 1, and ⊙ m intersects with X axis at two points a (2,0) and B (6,0) (1) Find the radius of ⊙ M; (2) Please judge the position relationship between ⊙ m and the line x = 7, and explain the reason

(1) Connect Ma,
∵ Mn ⊥ AB at point n,
∴AN=BN，
∵A（2，0），B（6，0），
∴AB=4，
∴AN=2；
In RT △ amn, Mn = 1, an = 2,
∴AM=
5，
That is, the radius of ⊙ m is
5；
(2) The line x = 7 is separated from ⊙ M,
Reason: the distance from the center m to the straight line x = 7 is 7-4 = 3,
∵3＞
5，
The straight line x = 7 is separated from ⊙ M