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In the plane rectangular coordinate system xoy, it is known that the circle m with m as the center passes through three points F1 (0, - C) F2 (0, c) a ((√ 3) C, 0), where C > 0 (1) finds the mark of circle M In the plane rectangular coordinate system xoy, it is known that the circle m with m as the center passes through three points F1 (0, - C) F2 (0, c) a ((√ 3) C, 0), where C > 0 (1) Find the standard equation of circle m (expressed by formula containing C) (2) It is known that the left and right vertices of the ellipse y ^ 2 / A ^ 2 + x ^ 2 / b ^ 2 = 1 (a > b > 0) (where a ^ 2-B ^ 2 = C ^ 2) are D, B, and the two intersections of circle m and X axis are a and C respectively, and point a is on the right side of point B, and point C is on the right side of point D. ① calculate the range of eccentricity of ellipse; ② if a, B, m, O, C, D, (o is the origin of coordinates) are uniformly distributed on the X axis in turn, Is the intersection of MF1 and df2 on a fixed line? If so, ask for the equation of the fixed line; if not, please explain the reason
(1) From F1 (0, - C) F2 (0, c), we can see that the center of the circle m is on the straight line y = 0, that is, on the X axis, let m (m, 0),
Because the distance from the center of the circle to the point on the circle is equal to the radius R
That is, MF1 = ma = R, so there is an equation: under the root sign {(M-0) ^ 2 + (0 + C) ^ 2} = under the root sign {(M - √ 3C) ^ 2 + (0-0) ^ 2} = R
M = the root of 3 times C [√ 3C / 3]
r=2√3c/3
r^2=4c^2/3
Standard equation (x-m) ^ 2 + y ^ 2 = R ^ 2
As shown in the figure, in the plane rectangular coordinate system, the straight line AB intersects two points a (6,0) and B (0,6), point C is a moving point on line AB, point P is on the line L: y = 3x-8, and the quadrilateral obcp is a parallelogram (1) Finding the analytic formula of line ab (2) Find the coordinates of point C (3) Can the parallelogram obcp be a diamond? Please explain the reason
y=6-x
Coordinates of point C (2, - 2)
If the distance between OP and ob is not equal, if it is diamond, the side length should be equal
The straight line 3x-4y-5 = 0 is cut by the circle x ^ 2 + y ^ 2 = 2, and the chord center angle of chord AB is?
1) The distance from the center of the circle to the straight line is the chord center distance
d=|0-0-5|/√(3^2+4^2)=1
2) According to the vertical diameter theorem, half of the chord length is calculated
Because: radius r = √ 2, chord center distance d = 1,
Therefore, half chord length = √ (R ^ 2-D ^ 2) = 1
So, the chord length is: 2
And because: radius r = √ 2
Therefore, according to the inverse theorem of Pythagorean theorem, the angle of chord center is 90 degrees
(2014. Quanzhou the first mock exam) if the line 3x-4y=0 intersects the circle x2+y2-4x+2y-7=0 at A and B two points, then the length of the string AB is equal to (). A. 2 B. 4 C. 2 Two D. 4 Two
The circle x2 + y2-4x + 2y-7 = 0 can be reduced to: (X-2) 2 + (y + 1) 2 = 12
The center coordinate of the circle is (2, - 1), and the radius is 2
3，
The distance from the center of the circle to the line 3x-4y = 0 is d = 6 + 4
5=2，
Therefore, the chord length | ab | = 2
12−4=4
2，
Therefore, D
The length of chord AB cut by the line 3x + 4y-15 = 0 by circle x2 + y2 = 25 is______ .
The center coordinate of x2 + y2 = 25 is (0, 0) and the radius is: 5, so the distance from the center of the circle to the straight line is: D = | - 15|
32+42＝3，
So 1
2|AB|=
52−32=4，
So | ab | = 8
So the answer is: 8
Find the length of chord ab of the line 3x-y-6 = 0 cut by circle C x + y-2x-4y = 0
The length of the chord AB cut by the circle C: x ^ 2 + y ^ 2-2x-4y = 0 is the length of the line L: 3x-6 = 0, which is obtained by the linear equation y = 3x-6, and then x ^ 2 + (3x-6) ^ 2-2x-4 (3x-6) = 0 is obtained by simplification, and x ^ 2-5x + 6 = 0 is obtained, and the solution is X1 = 2 x2 = 3. Therefore, Y1 = 0, y2 = 3, that is, the coordinates of the two intersections are (2,0); (3) the chord
In the plane rectangular coordinate system xoy, it is known that the line y = - 3 / 3 root sign 3x + 3 / 3 root sign 3 intersects X axis at point C and intersection Y axis at point a Let's rotate the vertex of the triangle o to the point of B so that the vertex of the triangle is equal to the point of B
According to the meaning of the title, the straight line y = - 3 / 3 root sign 3x + 2 / 3 root sign 3 intersects Y axis at point a, so point a is on the positive half axis of Y axis, so point B is in the first quadrant. OA = 2 / 3 root sign 3, because the vertex D of isosceles right triangle OBD (c) coincides with point C, OD = BD, and because ∠ OCA = 30 °, OC = od = 4, so the coordinates of point B are (4,4)
In the plane rectangular coordinate system xoy, let the straight line y = 3x + 2M and circle x2 + y2 = N2 are tangent, where m, n ∈ n, 0 ＜| M-N | ≤ 1, if the function f (x) = the zero point of MX + 1-N x0 ∈ (k, K + 1) k ∈ Z, then K=______ .
In the plane rectangular coordinate system xoy, the edge ab of rectangular ABCD is on the X axis, and ab = 3, BC = 2, root sign 3, the straight line y = root sign 3x-2, root sign 3 passes through point C and intersects the Y axis (3) Translate the parabola in (2) along the straight line y = root 3x-2 radical 3. After translation, the parabola intersects Y-axis at point F, and the vertex is e. whether there is such parabola after translation, is △ EFG an isosceles triangle? If so, request the parabola analytic formula at this time Concrete solution
Linear formula: y = √ 3x-2 √ 3, the edge ab of rectangular ABCD is on the X axis, and ab = 3, BC = 2 √ 3,
It can be concluded that the coordinates of point C are (4,2 √ 3), and a (1,0), B (4,0), D (1,2 √ 3) are obtained
What's your second parabola?
There are () A. 1 B. 2 C. Three D. Four
From the equation of the circle, the coordinates a of the center of the circle are (3,3), and the radius AE is 3,
Then the distance from the center of the circle (3, 3) to the line 3x + 4y-11 = 0 is d = | 3 × 3 + 4 × 3 − 11|
5 = 2, that is ad = 2,
ed = 1, that is, the distance between E on the circumference and the known straight line is 1, and the existence of P and Q also satisfies the question,
There are three points with a distance of 1 from the point on the circle to the line 3x + 4y-11 = 0
Therefore, C
RELATED INFORMATIONS
- 1. There are () A. 1 B. 2 C. Three D. Four
- 2. The line y = - 4 3x + 4, x-axis and y-axis intersect at point a and B respectively. In the plane rectangular coordinate system, the distance between a and B points to the straight line a is 2, then the number of lines a satisfying the condition is () A. 1 B. 2 C. 3 D. 4
- 3. In the plane rectangular coordinate system xoy, the intersection points of curve y = xsquare-4x + 3 and two coordinate axes are on circle C. find out the equation of circle C! 2, whether there is a real number a, so that the circle C and the straight line X-Y + a = 0 intersect at two points a and B, and the angle AOB = 90 ° is satisfied. If there is a value of a, if not, explain the reason
- 4. In the plane rectangular coordinate system xoy, the parabola y = ax ^ 2 + BX + C (a ≠ 0) passes through point a (- 3,0) and point B (1,0). Let the intersection point of parabola and Y axis be point C (1), directly write out the symmetry axis of parabola (2) and find the length of OC (expressed by algebraic formula of a)
- 5. As shown in Figure 1, in the plane rectangular coordinate system, the straight line AB intersects X axis at point a, intersects Y axis at point B, and point C is the moving point on line ab (1) If ∠ OAB is 20 ° larger than oba, OC ⊥ AB, calculate the degree of ∠ AOC; (2) As shown in Fig. 2, am bisection ∠ Bao, BM bisection ∠ OBN. When point a moves on the negative half axis of X axis, does the value of ∠ AMB change? If it does not change, calculate the degree of ∠ AMB; if it changes, please explain the reason; (3) As shown in Fig. 3, if ∠ OAB = 45 ° and ∠ OPA = ∠ BPD, ∠ BDP = ∠ ODF, then the following two conclusions are obtained: Please point out that there is only one correct conclusion
- 6. In the plane rectangular coordinate system xoy, there is an ellipse with F1 (0, - radical 3) and F2 (0, radical 3) as the focus and eccentricity as the root 3 / 2. Let the part of the ellipse in the first quadrant be a curve C, the moving point P is on C, the intersection points of tangent of C at point P and X, Y axis are a and B respectively, and vector om = vector OA + vector ob (1) The minimum value of locus equation (2) | vector om | of point m
- 7. In the plane rectangular coordinate system xoy, a line passing through the coordinate origin and the function f (x) = 2 If the image of X intersects P and Q, then the minimum PQ length of the line segment is______ .
- 8. In the plane rectangular coordinate system xoy, there are two intersections P and Q between the line L passing through the point (0, √ 2) and the ellipse x ^ 2 / 2 + y ^ 2 = 1 ① Find the value range of K ② Let the intersection points of ellipse and positive semiaxis of X axis be a and B respectively. Is there a constant k such that (vector OP + vector OQ) is collinear with vector AB? ② Supplementary condition: positive half axis with y axis
- 9. Given that the circle C: x + y-8y + 12 = 0, the straight line L passes through point d (- 2,0), and the slope is K (1). To solve the equation of circle e with line segment CD as its diameter, thank you, (2) If the line L is separated from the circle C, find the value range of K
- 10. The equation of a circle whose center is on the straight line X-Y-4 = 0 and passes through the intersection of two circles x ^ 2 + y ^ 2-4x-3 = 0, x ^ 2 + y ^ 2-4y-3 = 0? Circular system equation
- 11. As shown in the figure, in the plane rectangular coordinate system, the straight line y = 4 / 3x + 8 intersects the coordinate axis at two points a and B, AE bisection angle Bao intersects the Y axis at e, and point C is a point in the first quadrant of the line y = X Find the e-coordinate of the point, AE analytic formula problem supplement: chart
- 12. In the plane rectangular coordinate system, if the center of circle P (2, a) (a > 2), radius is 2, and the length of chord ab of circle P is two 3, then the value of a is I know the answer is 2 + radical 2, but why
- 13. As shown in Fig. 1, in the plane rectangular coordinate system, point a starts from (1,0) and moves along the x-axis at the speed of 1 unit length per second In the process of motion, the rhombus oabc is made with OA as one side, so that B and C are in the first quadrant and ∠ AOC = 60 ° to connect AC and ob. At the same time, point m starts from the origin O and moves along the diagonal line ob at the speed of three unit length of the root sign per second. If the circle is drawn with point m as the center and Ma as the radius, the movement time is set as T seconds 1. When t = 1, judge the position relationship between point O and circle m, and explain the reason 2. When the circle m is tangent to the OC edge, calculate the value of T 3. With the change of T, the number of common points on the four sides of circle m and diamond oabc is also changing. Please write down the corresponding relationship between the number of common points and the size of T
- 14. Point P is a point outside the line L, a, B, C are three points on the line L, PA = 4cm, Pb = 5cm, PC = 2cm, then the distance between point P and line L () A. Less than 2cm B. Equal to 2cm C. No more than 2cm D. It's 4 cm
- 15. If point P is a point outside the line L, ABC is three points above L, PA = 3cm, Pb = 4cm, PC = 5cm, then the distance from point P to line L is? A.3cm B.4cm C. Less than 3cm D. no more than 3cm
- 16. It is known that the radii of the intersecting circles are 5cm and 4cm respectively, and the common chord length is 6cm
- 17. If the radius of ⊙ o is 5cm and the length of chord AB is 6cm, then the distance from the midpoint of chord AB to the midpoint of inferior arc AB is______ .
- 18. What is the chord length of the straight line y = radical 3x cut by the circle x? + y? - 4Y = 0?
- 19. If the radius of ⊙ o is 5cm, and the shortest chord length passing through a point P in ⊙ o is 8cm, then Op=______ .
- 20. If the distance from a point on a straight line to the center of a circle is equal to the radius of the circle, then the straight line is the tangent line of the circle. B. the distance from the center of the circle to the straight line The correct one of the following propositions is () A. If the distance from a point on a line to the center of a circle is equal to the radius of the circle, the line is tangent to the circle B. If the distance from the center of the circle to the straight line is not equal to the radius, the line intersects the circle C. If a line and a circle have a unique common point, the line is tangent to the circle D. If the line AB has no intersection with the circle, then the line AB is separated from the circle PS: I feel both C and D are right~