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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
By substituting the string AB: y = K (X-2) into the ellipse
(9k^2+4)x^2-36k^2x+36(k^2-1)＝0.
Let the midpoint of a chord be (x, y), then we can get the result according to Weida's theorem
x＝(x1+x2)/2＝18k^2/(9k^2+4) …… (1)
y＝k(x-2)＝-8k/(9k^2+4) …… II.
When k = 0, the locus of the midpoint of chord AB is the point P itself, that is, x = 2
When k ≠ 0, the m-point trajectory can be obtained by eliminating the parameter K from ① and ②
(x-1)^2+y^2/(2/3)^2＝1.
Go over the circle O: x ^ 2 + y ^ 2 = 16. Make a straight line intersect the circle O at two points ab. find the locus of the midpoint C of the chord ab Can we use the point difference method to simplify the following 2x + 2Y * (y1-y2) / (x1-x2) = 0, how to bring the slope in? If there is no point difference method, is there any simple method other than Veda theorem? Can you tell me how to solve this type of problem faster?
The equation of the circle is x2 + y2-6x-8y = 0. If a chord of length 8 is made through the origin of coordinates, then the equation of the line where the chord is located is______ (general equation of the result written as a straight line)
X2 + y2-6x-8y = 0, i.e. (x-3) 2 + (y-4) 2 = 25. If the slope exists, let the straight line be y = KX
∵ the radius of the circle is 5, and the distance from the center m (3, 4) to the straight line is 3, ᙽ d = | 3K − 4|
k2+1=3，
∴9k2-24k+16=9（k2+1），∴k=7
24. The straight line is y = 7
24x；
When the slope does not exist, if the straight line is x = 0, verify that its chord length is 8, so x = 0 is also the line to be calculated. Therefore, the straight line is: x = 0 or 7x-24y = 0
So the answer is: x = 0 or 7x-24y = 0
The equation of the circle whose center is m is x2 + y2-6x-8y = 0. The straight line passing through the origin o of coordinates intersects the circle m at two points a and B, and the length of chord AB is 8
The equation of the circle can be expressed as (x - 3) 2 + (y-4) 2 = 25, the center of the circle m (3,4), the radius is 5, the slope of the line passing through the origin o is k, the equation is y = KX, KX - y = 0, the midpoint is C, AC = 8 / 2 = 4, AMC is a right triangle, MC = √ (MA? - AC?) = √ (
The equation of the circle is x2 + y2-6x-8y = 0. If a chord of length 8 is made through the origin of coordinates, then the equation of the line where the chord is located is______ (general equation of the result written as a straight line)
X2 + y2-6x-8y = 0, i.e. (x-3) 2 + (y-4) 2 = 25, if the slope exists, let the straight line be y = KX. ? the radius of the circle is 5, the distance from the center of the circle m (3, 4) to the straight line is 3, ? D = | 3K − 4 | K2 + 1 = 3, ᙨ 9k2-24k + 16 = 9 (K2 + 1), k = 724. ? the straight line is y = 724x
Given that the equation of the circle is x2 + y2-6x-8y = 0, if the longest chord and the shortest chord of the circle passing through the point (3,5) are AC and BD respectively, then the area of the quadrilateral ABCD is () A. 10 Six B. 20 Six C. 30 Six D. 40 Six
The standard equation of circle is (x-3) 2 + (y-4) 2 = 52,
The longest string | AC | = 2 × 5 = 10,
According to Pythagorean theorem, the shortest string | BD | = 2
52−12=4
6, and AC ⊥ BD,
The area of quadrilateral ABCD is s = | 1
2AC|•|BD|=1
2×10×4
6=20
6．
Therefore, B is selected
The equation of the circle is x ^ 2 + y ^ 2 + 8x-6y = 0. Find the chord with length of 8 through the coordinate origin, and find the line where the chord is Hurry up!
x^2+y^2+8x-6y=0
(x+4)^2+(y-3)^2=25
The center of the circle is (- 4,3) and the radius is 5
The line passing through the origin of coordinates can be set as y = KX
Chord length = 8 half of chord = 4 radius = 5
So the chord center distance = root sign (25-16) = 3
Distance from center of circle to straight line = | 4K + 3 | / under radical sign (k ^ 2 + 1) = chord center distance = 3
(4k+3)^2=9(k^2+1)
16k^2+24k+9=9k^2+9
7k^2+24k=0
k(7k+24)=0
K = 0 or K = - 24 / 7
So the line where the string is located is y = 0 or y = - 24x / 7
Find the equation of the circle passing through the origin and the two intersections of the circle x ^ 2 + y ^ 2 + 8x-6y + 21 = 0 and the straight line X-Y + 5 = 0
Join the circle x ^ 2 + y ^ 2 + 8x-6y + 21 = 0 and the line X-Y + 5 = 0
The intersection points a (- 2,3) and B (- 4,1) are obtained
Because the circle is over the origin, we should pay attention to the equation
Let x ^ 2 + y ^ 2 + CX + dy = 0
Substituting a and B, we get the equations of C and D
C = 19 / 5, d = - 9 / 5
So the equation is 5x ^ 2 + 5Y ^ 2 + 19x-9y = 0
Given that the circle C: x2 + y2-4x-12 = 0, AB is a chord of circle C, and the midpoint of AB is (3,1), then the equation of line AB is
Use the point difference method
Let a (x1, Y1), B (X2, Y2),
Then X1 ^ 2 + Y1 ^ 2-4x1-12 = 0,
x2^2+y2^2-4x2-12=0 ,
By subtracting the two formulas, we get (x2 + x1) (x2-x1) + (Y2 + Y1) (y2-y1) - 4 (x2-x1) = 0,
Since the midpoint of AB is (3,1), X1 + x2 = 6, Y1 + y2 = 2,
Substituting 6 (x2-x1) + 2 (y2-y1) - 4 (x2-x1) = 0,
It is found that K AB = (y2-y1) / (x2-x1) = - 1,
Therefore, the equation of AB is Y-1 = - (x-3), and X + y-4 = 0
If the circle x2 + y2-4x-5 = 0, then the equation of the line L where the shortest chord passing through point P (1,2) is () A. 3x+2y-7=0 B. 2x+y-4=0 C. x-2y-3=0 D. x-2y+3=0
If the chord is the shortest, the line between the center of the circle and the point P is perpendicular to the line L
The center of the circle is O (2, 0)
∴Kl＝ −1
KOP＝1
Two
The linear equation obtained by the point oblique formula is as follows:
x-2y+3=0
Therefore, D is selected
RELATED INFORMATIONS
- 1. 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
- 2. 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)
- 3. 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
- 4. 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
- 5. 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
- 6. 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
- 7. It is known that AC is the diameter of circle 0, PA is vertical AC, connecting OP, xuancb is parallel OP, the straight line Pb intersects the straight line AC at point D, BD = 2PA
- 8. P is a point outside the circle O, Pb is the tangent line of circle O, a and B are the tangent points, BC is the diameter of circle O, so AC is parallel to Po As the title
- 9. Given that the circle C (x + radical 3) ^ 2 + y ^ 2 = 16, point a (radical 3,0) q is a moving point on the circle, the vertical bisector of AQ intersects CQ at point m, and the trajectory equation of M is e Find the equation of E. urgent! The area of the l-intersection locus e of the straight line passing through the point P (1,0) at two different points a, B and △ AOB is 4 / 5
- 10. As shown in the figure, AB is the diameter of ⊙ o, C is ⊙ o, and the tangent line passing through point C is perpendicular to each other, and the perpendicular foot is D, so as to verify the AC bisection angle DAB As shown in the figure, AB is the diameter of ⊙ o, C is the point on ⊙ o, CD cuts ⊙ o at point C, [label: ab CD, diameter, CD]
- 11. 0
- 12. (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
- 13. 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
- 14. 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
- 15. 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
- 16. 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
- 17. The common length of the two circles obtained by + 1y + 10Y + 2x = 0-2
- 18. 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
- 19. 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
- 20. The ratio of () to () of a circle is called PI, which is represented by the letter ()