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What is the chord length of a line passing through the origin and with an inclination angle of 60 degrees cut by the circle x square + y square - 4Y = o?
The standard equation of a circle is: x ^ 2 + (Y-2) ^ 2 = 4
So the coordinates of the center of the circle are (0,2), and the radius is r = 2
The linear equation is y = radical 3 * X
Let the distance from the center of the circle to the straight line be D, d = 1,
r^2=d^2+(l/2)^2
L = 2 * radical 3
The equation of a circle passing through the intersection of the straight line 2x + y + 4 = 0 and the circle x2 + Y2 + 2x-4y + 1 = 0 and satisfying one of the following conditions: (1) Crossing the origin (2) There is a minimum area
The equation of the circle passing through the intersection point of the straight line 2x + y + 4 = 0 and the circle x2 + Y2 + 2x-4y + 1 = 0 can be set as (x2 + Y2 + 2x-4y + 1) + λ (2x + y + 4) = 0 (1). By substituting (0, 0), we can get 1 + 4 λ = 0, λ = - 14, the equation of circle is x2 + Y2 + 32x − 174y = 0; (2) (x2 + Y2 + 2x-4y +... "
Given that the line passing through the point (0,1) intersects the circle x square + y square - 2x + 4Y = 0, and the chord length is 4, then the equation of the line is
Circle x square + y square - 2x + 4Y = 0
(x-1)^2+(y+2)^2=5
Center coordinates (1, - 2) r = √ 5
And the slope of the straight line is k, then the equation is y = KX + 1
Half chord m, distance d from center of circle to straight line, radius r constitute Pythagorean theorem
m=2 r=√5 d=√(r^2-m^2)=1
d=|k+3|/√(k^2+1)
So | K + 3 | / √ (k ^ 2 + 1) = 1
k^2+6k+9=k^2+1
k=-4/3
So the equation is y = - 4 / 3x + 1
In addition, the distance between the center of a circle and its center is equal to 1
So x = 0 is the same
The equation of a straight line with chord length of root 3 obtained from the circle x ^ 2 + y ^ 2-2x = 0 is
Circle x ^ 2 + y ^ 2-2x = 0
（x-1）²+y²=1
When the length of the cut chord is root 3, the center of the circle goes to the straight line
The distance is 1 / 2,
d=|k|/√(k²+1)=1/2
The solution is k = ± √ 3 / 3
The equation of a straight line is y = = ±√ 3 / 3x
The chord length of circle x2 + y2-4x + 4Y + 6 = 0 is equal to () A. Six B. 5 Two Two C. 1 D. 5
Given that the circle x2 + y2-4x + 4Y + 6 = 0, the center of the circle is (2, - 2), and the radius is
2．
The center of the circle is (2, - 2) to the straight line x-y-5 = 0
Two
2．
Using geometric properties, the chord length is 2
(
2)2−(
Two
2)2=
6．
Therefore, a
The chord length of the straight line X-Y + 2 = 0 cut by the circle x? 2 + y? 2 + 4x-4y-8 = 0 is equal to?
First, the equation of a circle is transformed into a general equation, i.e. (x + 2) 2 + (Y-2) 2 = 16
The chord length of the circle x ^ 2 + y ^ 2-4x + 4Y + 4 = 0 cut by the line x-y-5 = 0 is equal to
Circle equation: (X-2) ^ 2 + (y + 2) ^ 2 = 4
Center O (2, - 2), radius r = 2
The distance from O to the line is d = |2 + 2-5| / Radix 2 = Radix 2 / 2
According to Pythagorean theorem, half of chord length = root [R ^ 2-D ^ 2] = root (4-1 / 2) = root 14 / 2
That is, chord length = root 14
Find the length of chord cut by the line L: x-y-5 = 0 for circle C: x 2 + y 2-4x + 4Y + 4 = 0
The circle C: x2 + y2-4x + 4Y + 4 = 0 can be reduced to (X-2) 2 + (y + 2) 2 = 4,
The center coordinate of the circle is C (2, - 2), and the radius of the circle is 2,
| the distance from the center of the circle C to the line L: x-y-5 = 0 is | 2 + 2 − 5|
2=
Two
2，
The chord length of circle C: x2 + y2-4x + 4Y + 4 = 0 cut by the line L: x-y-5 = 0 is 2
4−1
2=
14．
The chord length of circle x2 + y2-4x + 4Y + 6 = 0 is equal to () A. Six B. 5 Two Two C. 1 D. 5
Given that the circle x2 + y2-4x + 4Y + 6 = 0, the center of the circle is (2, - 2), and the radius is
2．
The center of the circle is (2, - 2) to the straight line x-y-5 = 0
Two
2．
Using geometric properties, the chord length is 2
(
2)2−(
Two
2)2=
6．
Therefore, a
If the chord length of the straight line 2aX by + 2 = 0 (a > 0, b > 0) cut by circle x2 + Y2 + 2x-4y + 1 = 0 is 4, then 1 A+1 The minimum value of B is______ .
The circle x2 + Y2 + 2x-4y + 1 = 0, i.e. (x + 1) 2 + (Y-2) 2 = 4, the center of the circle is (- 1, 2), and the radius is 2,
If the distance from the center of a circle to the straight line 2aX by + 2 = 0 is equal to D, then 2 is obtained from the chord length formula
4 − D2 = 4, d = 0, i.e
The straight line 2aX by + 2 = 0 passes through the center of the circle, ﹥ 2a-2b + 2 = 0, a + B = 1,
Then 1
A+1
b=a+b
a+a+b
b=2+b
A+a
b≥2+2
B
a•a
If and only if a = B, the equal sign holds,
So the minimum value of the equation is 4, so the answer is 4
RELATED INFORMATIONS
- 1. 2 ax by + 2 = 0 (a > 0, b > 0). The chord length cut by the circle x ^ 2 + y ^ 2 + 2x + - 4Y + 1 = 0 is 4
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