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It is known, as shown in the figure: in the plane rectangular coordinate system, O is the coordinate origin, the quadrilateral oabc is a rectangle, the coordinates of points a and C are a (10, 0), C (0, 4), respectively; point D is the midpoint of OA, and point P moves on the edge of BC; when △ ODP is an isosceles triangle with waist length of 5, the coordinates of point P are______ .
(1) When od is the bottom of an isosceles triangle, P is the intersection of the vertical bisector of OD and CB, and op = PD ≠ 5;
(2) When od is one waist of isosceles triangle:
① If point O is the vertex of the vertex angle, point P is the intersection point of the arc with point o as the center of the circle and radius of 5 with CB,
In right angle △ OPC, CP=
OP2-OC2=
If 52-42 = 3, then the coordinates of P are (3, 4)
② If D is the vertex of the vertex angle, point P is the intersection point of the arc with point D as the center of the circle and radius of 5 with CB,
D as DM ⊥ BC at point M,
In right angle △ PDM, PM=
PD2-DM2=3，
When p is on the left of M, CP = 5-3 = 2, then the coordinates of P are (2,4);
When p is on the right side of M, CP = 5 + 3 = 8, then the coordinates of P are (8, 4)
So the coordinates of P are: (3,4) or (2,4) or (8,4)
So the answer is: (3,4) or (2,4) or (8,4)
In the rectangular oabc, O is the origin of plane rectangular coordinate system, the coordinates of points a and C are (3,0), (0,5), and point B is in the first quadrant (1) Write the coordinates of point B (,) (2) If the straight line CD passing through point C intersects AB with point D, and the circumference of the rectangular oabc is divided into two parts, i.e. 3:1, to find the coordinates of point D (3) The area of the quadrilateral oad1c1 is calculated by translating the line CD in (2) down two unit lengths to obtain C 1D 1
(1) Because CB / / OA, AB / / OC, so B (3,5) (2) know that the circumference of the rectangle is 3 * 2 + 5 * 2 = 16, so 3:1 = 12:4. Because the straight line passes through C and D is on AB, CB + BD = 4, CB = OA = 3, so BD is equal to 1, so D ordinate is 4-1 = 3, that is, D point coordinate is (3,3) (3) because c1d1 is CD down translation two
It is known that the radius of the circle O is r, the distance from the center of the circle O to the line L is D, and D and R are the two roots of the equation x 2 + 2 MX + 1 = 0, Given that the radius of the circle O is r, the distance from the center of the circle O to the straight line L is D, and D and R are the two roots of the equation x? + 2mx + 1 = 0, then when the line L is tangent to ⊙ o, the value of M is
If a line is tangent to a circle, then d = R
So the equation has equal roots
Then (2m) ^ 2-4 = 0
M = 1 or - 1,
However, when m = 1, two are negative roots, and they are discarded
So we can only take M = - 1
Given that the radius of two circles is r1r2, they are two of the equations 2x? - 14x + 5 = 0, and the distance between the centers of two circles is 7, what is the position relationship between the two circles?
∵ the radius of two circles is R1, R2
Moreover, R1 and R2 are two of the equations 2x? 14x + 5 = 0
from Weida's theorem, it can be concluded that:
R1+R2=14/2=7
∵ center distance = 7
ν R1 + R2 = center distance of circle
The two circles are circumscribed
Given that the equation of the straight line L is 3x + 4y-25 = 0, then the minimum distance between the point on the circle x2 + y2 = 1 and the line L is () A. 3 B. 4 C. 5 D. 6
∵x2+y2=1
The center of the circle (0, 0), with a radius of 1
The distance from the center of the circle to the straight line is: D = 25
32+42＝5
As shown in the figure: the minimum distance between the point on the circle x2 + y2 = 1 and the straight line L is D-R = 4
Therefore, B is selected
Find the equation of the circle passing through the intersection of two circles x 2 + y 2 - 2 x - 2 y + 1 = 0 and x 2 + y 2 - 6 x - 4 y + 9 = 0, and the center of the circle is on the straight line y = 2x
According to the meaning of the question, let the equation of the circle be x? + y? - 2x-2y + 1 + λ (x? + y? - 6x-4y + 9) = 0, and λ is an unknown number. Because the center of the circle is on the straight line y = 2x, the coefficient in front of the Y term of the circular equation must be twice that of the coefficient in front of the X term
Two parallel straight lines 3x + 2y-6 = 0 and 6x + 4y-3 = 0 are known, and the equations of parallel lines equidistant from them are solved
Method 1: let the straight line be 6x + 4Y + T = 0, turn 3x + 2y-6 = 0 into 6x + 4y-12 = 0, then | T + 12 | / √ 6 | 4 | = | T + 3 | / √ 6 ﹤ 4 | t = - 15 / 2, so the equation is 6X + 4y-15 / 2 = 0
Find the length of the string AB that the straight line 3x minus 4Y plus five equals zero and the circle x square plus y square equals to four
It is known that the radius of the circle is 2 and the distance from the center of the circle to the straight line is 1. Using the Pythagorean theorem, it is concluded that half of the chord is root 3, so the chord length is 2 * root 3
We can also combine the equations to get the intersection point and find the chord length
You can also use the string length formula. However, this problem is relatively simple, and the above two methods are faster
Find the linear equation of the longest chord and the shortest chord passing through a point P (- 5, - 1) in the circle x square + y square + 6x-4y-3 = 0
The circle x ^ 2 + y ^ 2 + 6x-4y-3 = 0, i.e. (x + 3) ^ 2 + (Y-2) ^ 2 = 10 center C (- 3,2), radius √ 10, the shortest chord passing through the point P (- 5, - 1) of the circle is the chord with P as the midpoint. The product of the vertical slope k and the slope of PC is - 1kpc = (2 + 1) / (- 3 + 5) = 3 / 2 ﹤ k = - 2 / 3
Find the common chord length of circle x square plus y square minus four equal to zero and x square plus y square minus four x plus 4Y minus 12!
The first circle: Center (0,0) radius = 2 second circle: Center (2, - 2) radius = 2. Consider the line between one of the intersection points of two circles and the line between the centers of two circles. The length of the triangle is 2,2,2 * (2) ^ 0.5, which indicates that the three sides form an isosceles right triangle
RELATED INFORMATIONS
- 1. It is known that the straight line L is tangent to the circle C: x square + y square + 2x-4y + 4 = 0, and the distance from the origin o to L is 1, and the equation of the straight line L is obtained
- 2. (2006 Yantai) known: on the one variable quadratic equation X2 - (R + R) x + 1 4D2 = 0 has no real root, where R and R are the radii of ⊙ O1, ⊙ O2 respectively, and D is the center distance between the two circles, then the position relationship of ⊙ O1, ⊙ O2 is () A. Exotropism B. Tangency C. Intersection D. Contains
- 3. It is known that a vertex of an ellipse is a (0, - 1) and the focus is on the x-axis. If the distance between the right focus and the root of the line X-Y + 2 is 2 = 0, the distance is 3 Let the ellipse intersect with the straight line y = KX + m and have different two points m n when am = an to find the value range of M
- 4. How many points on the circle x ＾ 2 + y ＾ 2 + 2x + 4y-3 = 0 to the straight line x ＾ y + 1 = 0 is the root sign 2! Although there are online solutions, but I still don't understand! Please explain!
- 5. 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~
- 6. If the radius of ⊙ o is 5cm, and the shortest chord length passing through a point P in ⊙ o is 8cm, then Op=______ .
- 7. What is the chord length of the straight line y = radical 3x cut by the circle x? + y? - 4Y = 0?
- 8. 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______ .
- 9. It is known that the radii of the intersecting circles are 5cm and 4cm respectively, and the common chord length is 6cm
- 10. 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
- 11. In the plane rectangular coordinate system, the points a (- 4,0) and B (2,0) are known. If the point C is in the first order function y = - 1 On the image of 2x + 2, and △ ABC is a right triangle, then the point C satisfying the condition has () A. 1 B. 2 C. Three D. Four
- 12. In the plane rectangular coordinate system, O is the coordinate origin. The point P (m, n) is on the image of the inverse scale function y = K / X (1) If M = k, n = K-2, find the value of K (2) If M + n = radical 2 K, Op = 2, and the inverse proportional function y = K / X satisfies: when x > 0, y decreases with the increase of X, and the value of K is calculated
- 13. As shown in the figure, in the plane rectangular coordinate system, the image of quadratic function intersects C at Y axis, a and B at x axis, points a and B are on both sides of the origin The coordinate of a is (- 3,0), Ao = co = 3BO (1) Find this quadratic function expression (2) If the point d (- 2, y) is a point on the parabola, find the area of △ BCD (3) In (2), if point P is a moving point between B and D on the parabola, when P moves to what position, the area of △ BDP is the largest? Find the coordinates of point P and the maximum area of △ BDP!
- 14. As shown in the figure, in the plane rectangular coordinate system, point O is the origin, the diagonal ob of diamond oabc is on the X axis, and the vertex A is in the inverse scale function y = 2 On the image of X, the area of the diamond is______ .
- 15. It is known that O is the origin of the plane rectangular coordinate system. The straight line passing through the point m (- 2,0) and the circle x ^ 2 + y ^ 2 = 1 intersect P and Q If the vector OP * vector OQ = - 1 / 2, find the equation of the line L If the area of the triangle OMP and the triangle OPQ are equal, find the slope of the line L
- 16. In the plane rectangular coordinate system, the coordinate of point a is (4,0), the point P is on the straight line y = - x + m, and AP = OP = 4
- 17. As shown in the figure, it is known that in the rectangular coordinate system xoy, points a and C are on the positive half axis of the x-axis and y-axis respectively, and the point O is at the coordinate origin. The right angle vertex o of the isosceles right angle triangle OEF is at the origin, e and F are on OA and OC respectively, and OA = 4, OE= 2. Rotate the triangle plate OEF anticlockwise around point O to oe1f1, and connect CF1 and AE1 (1) Verification: △ oae1 ≌ △ ocf1; (2) If the triangle plate OEF rotates anticlockwise around point O, is there a position that makes OE ∥ CF? If yes, request the coordinates of e point; if not, please explain the reason
- 18. As shown in the figure, in the plane rectangular coordinate system, the coordinates of the three vertices of △ ABC are a (2, 3), B (2, 1), C (3, 2) (1) Judge the shape of △ ABC; (2) If △ ABC is rotated one circle along the line of edge AC, the volume of the rotating body is obtained
- 19. In a rectangular coordinate system, a point whose abscissa and ordinate are integers is called a lattice point. Given that the center of a circle is at the origin and its radius is equal to 5, then the grid point on the circle has______ One
- 20. Given that the length of a chord is equal to the radius r, find: (1) The length of the inferior arc to which the string is directed; (2) The area of the bow formed by this chord and minor arc