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Finding the minimum area of the intersection of the two circles of the sum of the two Y 2-2x-2x-2y = 0-2x-2
To solve the equations
x2+y2-2x-2y-2=0
x2+y2-4x-4y=0
Results: x = (1 + √ 7) / 2, y = (1 - √ 7) / 2, or x = (1 - √ 7) / 2, y = (1 + √ 7) / 2,
The two intersections: a ([1 + √ 7] / 2, [1 - √ 7] / 2), B ([1 - √ 7] / 2, [1 + √ 7] / 2),
AB^2=(√7)^2+(√7)^2=14
AB midpoint: (1 / 2,1 / 2),
The smallest circle is a circle whose diameter is ab
∴(X-1/2)^2+(Y-1/2)^2=7/2.
Given the equation of circle x2 + y2-6x-6y + 14 = 0, find the trajectory equation of midpoint m of chord PQ of straight line crossing point a (- 3, - 5)
M(x,y)
C:x^2+y^2-6x-6y+14=0
C(3,3)
k(AM)*k(CM)=-1
[(y+5)/(x+3)]*[(y-3)/(x-3)=-1
x^2+(y+1)^2=25
Let the straight line x + 2Y + 4 = 0 and X2 + y2-2x-15 = 0 intersect at a, B (to find the vertical bisector equation of chord AB) and (to find the length of chord AB)
(1) The formula of the circle equation shows that: (x-1) 2 + y 2 = 16, then the coordinates of the center of the circle are (1,0), and the radius r = 4
Given that the straight line x + 2Y + 4 = 0 and the circle intersect at point A. B, we can know from the vertical diameter theorem that:
The vertical bisector of string AB must pass through the center (1,0)
If the slope of the straight line AB is - 1 / 2, the slope of its vertical bisector is 2
Therefore, from the point oblique equation of the straight line, it can be obtained that:
The equation of the vertical bisector of string AB is y = 2 (x-1), that is, 2x-y-2 = 0
(2) Let the length of chord AB be L
From (1), we can get the distance from the center of the circle (1,0) to the straight line AB: x + 2Y + 4 = 0
d=|1+4|/√5=√5
Because R 2 = D 2 + (L / 2) 2, so:
(L/2)²=r²-d²=16-5=11
The solution is L = 2 √ 11
So the length of string AB is 2 √ 11
The length of the chord AB cut by the circle C: x2 + y2-2x-4y = 0 of the straight line L passing through the point P (4, - 4) is 8. Find the equation of the straight line L. (x2 represents the square of x)
Circle C: x 2 + y 2 - 2x-4y = 20
（x-1）²+（y-2）²=25
Center (1,2), radius = 5
Chord length = 6, according to Pythagorean theorem
Calculate the distance from the center of the circle to the straight line = 3
Let the linear equation: y + 4 = K (x-4)
kx-y-4k-4=0
According to the formula of distance from point to line
｜k-2-4k-4｜/√（k²+1）=3
｜k+2｜=√（k²+1）
k²+4k+4=k²+1
4k=-3
k=-3/4
The equation of the line is - 3 / 4x-y + 3-4 = 0, that is, 3x + 4Y + 4 = 0
The other line is x = 4, and the slope does not exist, that is, the line is tangent to the circle C
In the straight line passing through point (2,1), the equation of the shortest chord length of the cut circle x2 + y2-2x + 4Y = 0 is obtained What is the shortest? Write down the answer What is the shortest case? Write down the answers and the detailed process I don't know how to find the shortest string outside the circle? The distance from the point to the chord is root 10, which is far greater than radius root 5. How to make a chord? The longest chord is the radius. What is the shortest chord?
The original formula = (x-1) ^ 2 + (Y-2)) ^ 2 = 5
The center of the circle (1, - 2) | R | = √ 5
Substituting (2,1), (1, - 2) into y = KX + B
-2=k+b
1=2k+b
K = 3, B = - 5
∴y=3x-5
A straight line perpendicular to y = 3x-5
y=-1/3x+b
Substituting (2,1) into
b=5/3
∴x+3y-5=0
Let the locus of the midpoint of the chord intercepted by the circle x * 2 + y * 2 = 2 for the straight line x + ky-1 = 0 be m, then the position relationship between the curve and the straight line x-y-1 = 0 is?
As shown in the figure, the straight line x + ky-1 = 0 passes the fixed point a (1,0),
From the knowledge of plane geometry, OM ⊥ am,
Thus, the locus of the midpoint m is a circle with the diameter of OA,
The equation is: (x - 12) 2 + y2 = 14,
From the equation of the circle, the coordinates of the center of the circle (12,0) and the radius r = 1 are obtained
Then the distance from the center of the circle (12,0) to the straight line x-y-1 = 0 is d = 125 < R = 12,
So the position relationship between the line and the circle is intersection
Therefore, C
The trajectory equation of the midpoint of a chord cut by the square of hyperbola X - Y / 4 = 1 I know how to work out the equations, but there are also constraints, I don't know how to calculate the constraints!
The coordinates of the two intersection points of the chord and hyperbola are: (x1, Y1), (x1, Y2) where the coordinates of the point coordinate is (x, y) where the point coordinate is (x, y) then there are: 2x = X1 + x2,2y = Y1 + y2x1 ^ 2-y11 ^ 2 / 4 = 1. [1] x2 ^ 2-y2 ^ 2 / 4 = 1. [2] [1] [1] - [2]: (x1 ^ 2-x2 ^ 2) = Y1 ^ 2 / 4-y2 ^ 2 / 4 (x1 + x2) (x1 + x2) (x1-x2) = (Y1 + Y2) (y1-y2) / 42x (x1-y1-y2) / 42x (x1-y1-y2) / 42x (x1-y1-y2) / 42x (x1-x1-y2) / 42x (x1 x2) = 2Y (Y1
Given that the radius of the circle is 2, the center of the circle is on the positive half axis of the X axis, and the circle is tangent to the straight line 3x + 4Y + 4 = 0, then the standard equation of the circle is______ .
Let the coordinates of the circle center be (a, 0) and a > 0,
Because the circle is tangent to the line 3x + 4Y + 4 = 0, the distance from the center of the circle to the line is equal to radius 2, that is, | 3A + 4|
32 + 42 = 2, a = 2 or a = - 14
3, so a = 2
The standard equation of the circle with the center coordinate of (2,0) and radius of 2 is: (X-2) 2 + y2 = 4
So the answer is (X-2) 2 + y2 = 4
Given that the circle C passes through the point (2,1) and the center of the circle is on the x-axis, the straight line L: 3x + 4Y-2 = 0 is tangent to the circle, and the equation of the circle is solved
Let the center of the circle be (a, 0) and the radius be r
Then the circle equation is (x-a) ^ 2 + y ^ 2 = R ^ 2
Because the circle passes through the point (2,1)
So (2-A) ^ 2 + 1 ^ 2 = R ^ 2
It is tangent to the line 3x + 4Y-2 = 0
Then | 3a-2 | / √ (3 ^ 2 + 4 ^ 2) = R. ②
A = 11 / 4, r = 5 / 4
So the equation for the circle is (X-11 / 4) ^ 2 + y ^ 2 = 25 / 16
If you do not understand, please hi me, I wish you a happy study!
Find the equation of circle whose chord length is 6 on the line L1: x-y-1 = 0 and tangent to the line l2:4x + 3Y + 14 = O, and cut on the line l3:3x + 4Y + 10 = 0
First of all, if the center of the circle is on the line L1: x-y-1 = 0, then we may as well set the coordinates of the center of the circle as (a, A-1). Because the circle is tangent to the line l2:4x + 3Y + 14 = 0, then the radius r = D = [4A + 3 (A-1) + 14] / 5 = (7a + 11) / 5. The chord length of the circle cut on the line l3:3x + 4Y + 10 = 0 is 6
RELATED INFORMATIONS
- 1. It is known that the center of circle C is tangent to the line l2:4x + 3Y + 14 = 0 on the line L1: x-y-1 = 0, and the straight line l3:3x + 4Y + 10 = 0 is obtained. The chord length is 6, and the equation of circle C is obtained
- 2. (1) Find the equation of straight line passing through the intersection point of the line l1:7x-8y-1 = 0 and l2:2x + 17Y + 9 = 0 and perpendicular to the line 2x-y + 7 = 0 (2) The straight line L passes through the point P (5, 5) and intersects the circle C: x2 + y2 = 25, and the chord length is 4 5. Find the equation of L
- 3. 0
- 4. Through a point P (2,0) in the ellipse x ^ 2 / 9 + y ^ 2 / 4 = 1, the chord AB is induced and the trajectory equation of the midpoint m of the chord is obtained Thinking
- 5. Given that the straight line passes through the point m (2,1), and the chord length cut by the circle C: (x-1) 2 + (y + 1) 2 = 4, the chord length is 2 √ 2, and the equation of the straight line is obtained
- 6. If the circle x2 + Y2 + 2x-4y + 1 = 0 is symmetric about the straight line 2aX by + 2 = 0 (a, B ∈ R), then the value range of AB is () A. (−∞，1 4] B. [1 4，+∞) C. (−1 4，0) D. (0，1 4)
- 7. If the common chord length of circle x2 + y2 = 4 and circle x2 + Y2 + 2ay-6 = 0 (a > 0) is 2 3, then a is equal to () A. 1 B. Two C. Three D. 2
- 8. It is known that the length of the two chords of a circle AB CD is two of the equation x ^ 2-42x + 432 = 0, and ab is parallel to CD, and the distance between the two chords is 3. Find the radius
- 9. As shown in the figure, AB and CD are the two parallel chords in the circle O, M is the midpoint of AB, and the extension line of DM intersects the circle O at e. it is proved that O, m, e and C are in the same circle
- 10. As shown in the figure, the radius of ⊙ o is 1cm, and the lengths of strings AB and CD are respectively 2 cm, 1 cm, then the acute angle α between the strings AC and BD=______ Degree
- 11. If P9 (2, - 1) is the midpoint of the chord ab of the circle (x-1) 2 + y2 = 25, what is the equation of AB
- 12. The common chord length of circle x2 + y2-4 = 0 and circle x2 + y2-4x + 4y-12 = 0 is () A. Two B. Three C. 2 Two D. 3 Two
- 13. The common length of the two circles obtained by + 1y + 10Y + 2x = 0-2
- 14. C 1: x 2 + y 2-2x-6y-6 = 0, and the number of common tangents of circle C2: x2 + y2-4x + 2Y + 4 = 0
- 15. The position relation of two circles C1: x2 + Y2 + 2x-6y-26 = 0 C2: x2 + y2-4x + 2Y = 0 I'm sorry the title is like this. I just missed the position relationship between two circles C1: x2 + Y2 + 2x-6y-26 = 0 C2: x2 + y2-4x + 2y-4 = 0
- 16. The ratio of () to () of a circle is called PI, which is represented by the letter ()
- 17. In order to know that the circumference of a square is equal to that of a circle, the side length of a square is 6.28 cm, and the circumference of the circle is () cm? The area is () square centimeter?
- 18. Draw the largest circle in the square. The circumference of the circle is 12.56 cm. The area of the square is______ .
- 19. Draw the largest circle in a square with a side length of 6cm. The circumference of the circle is () cm and the area is () square centimeter
- 20. Relating to the PI and the circumference of a circle The circumference of a circle = Pi times d or 2 times R. are these two formulas? Are they used to calculate the circumference? If so, then I understand the first formula It's PI times diameter The second one doesn't understand 2 times the radius? 2 times the radius. I understand. It's the diameter and then the circumference But what does the multiplication radius of the 2-fold school represent? What is the multiplication of two 3.14? What does it represent? In addition, what does the PI represent? What is the relationship between PI and radius, diameter and circumference? Please give an example to illustrate What does circumference apart from diameter represent?