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Given that the equation of a circle is X & sup2; + Y & sup2; - 4x + 4Y + 4 = 0, then the shortest distance from the point on the circle to the origin is and the longest distance is
x²-4x+4+y²+4y+4=4
(x-2)²+(y+2)²=4
Represents a circle whose center (2, - 2) radius is 2
Combination of number and shape
The distance from the center of the circle to the origin = √ (2-0) &# 178; + (- 2-0) &# 178; = 2 √ 2
So the shortest distance = 2 √ 2-2, the longest distance = 2 + 2 √ 2
As shown in the figure, the intersection circle x2 + (Y-1) 2 = 1 of the moving line passing through the origin is at point Q. take point P on the line OQ so that the distance from P to the line y = 2 is equal to | PQ | and solve the trajectory equation of point P when the moving line rotates around the origin
P (x, y, P (x, y), O1: x2 + (Y-1) 2 = 1 and y = 2, circle O1: x2 + (Y-1) 2 = 1 and line y = 2, tangent to point a, connect AQ, easy to know |aq ? = |||||||||||aq | = |||\\\\124\124\\\| ar = \124\\124\\\\\\\\x2 + y 2 = 4 is the orbit equation
RELATED INFORMATIONS
- 1. It is known that the equation of circle C is (x-a) 2 + (Y-A + 1) 2 = 1. When the distance between the origin and the center of circle C is the smallest, the value of a is obtained
- 2. Given the circle C: (x-1) 2 + (y + 1) 2 = 1, then the distance from the center of the circle C to the origin of the coordinate o is___ ．
- 3. Given that the point P (1,2) is on the circle C: X & # 178; + Y & # 178; - 2kx-2y + K & # 178; = 0, then the positional relationship between O (0,0) and circle C is
- 4. If two tangents of circle (x-1) × (x-1) + y × y = 1 can be made through points (5a + 1,12a), then the value range of real number a is
- 5. If all the points on the curve C: x2 + Y2 + 2ax-4ay + 5a2-4 = 0 are in the second quadrant, then the value range of a is______ ．
- 6. Given the circle m; 2x * x + 2Y * y-8x-8y-1 = 0 and the straight line L: x + Y-9 = 0, make a triangle ABC through a point a on the straight line L to make the angle BAC = 45 degrees, AB passes through the center m, and B, C are on the circle M 1: When the abscissa of a is 4, the equation of line AC is obtained 2: Find the value range of abscissa of point a
- 7. Given that the line L: x + Y-9 = 0 and the circle M: 2x & sup2; + 2Y & sup2; - 8x-8y-1 = 0, the point a (4, Y0) is on the line L, B and C are two points on the circle M In the triangle ABC, ∠ BCA = 45 ° AB passes through the center of circle m, then the distance from the center of circle m to the straight line AC is
- 8. Given that the point m is on the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1, the circle centered on M is tangent to the X axis and the right focus F of the ellipse, if the circle m is compared with the Y axis A. B two points, and the triangle ABM is an equilateral triangle with side length 2, the equation for solving ellipse
- 9. Find the linear equation of the longest and shortest chord of a point P (3,1) in the circle X & sup2; + Y & sup2; - 8x-2y + 8 = 0
- 10. The straight line passing through the point P (- 2,3) is cut by the circle X & sup2; + Y & sup2; - 4x + 2y-2 = 0, and the linear equation of the longest chord is obtained
- 11. What is the distance between the center of the circle x ^ 2 + y ^ 2-4x + 2Y + 1 = 0 and the straight line X-Y + 1 = 0
- 12. If there are only three points on the circle (x-1) ^ 2 + (y + 1) ^ 2 = 4 and the distance from them to the straight line 4x + 3Y = m is 1, then the value of M is?
- 13. If circle: x2 + y2-4x + 6y = 0 and circle: x2 + y2-6x = 0 intersect a and B, then the equation of the vertical bisector of AB is______ ．
- 14. If circle: x2 + y2-4x + 6y = 0 and circle: x2 + y2-6x = 0 intersect a and B, then the equation of the vertical bisector of AB is______ ．
- 15. The equation of the circle symmetric with the circle X & sup2; + Y & sup2; - 2x-6y + 9 = 0 about the line x-y-1 = 0
- 16. Given that f (x) = 2x ^ 2-6x + a (a is a constant), there is a maximum value of 3 on [- 2,2]. Find the minimum value of F (x) on [- 2,2]
- 17. Let y = 2 (the square of cosx) - 2acosx - (2a + 1) minimum value f (a) of X function, find the analytic expression of F (a), and find the method
- 18. Let the minimum value of the function y = 2 (cosx) 2-2acosx - (2a + 1) about X be 0.5, and find the maximum value
- 19. Given the function y = X2, - 2 ≤ x ≤ a, where a ≥ - 2, find the maximum and minimum value of the function, and find the value of the independent variable x corresponding to the maximum and minimum value of the function
- 20. Given that the coordinate of point a is (0,1), the coordinate of point B is (2 / 3, - 2), and point P moves on the line y = - x, when the absolute value of PA minus Pb is the maximum, the coordinate of point P is?