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(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
According to the meaning of the question, if the equation has no real root, we can get (R + R) 2-d2 < 0,
Then: (R + R + D) (R + R-D) < 0,
Because R + R + d > 0, R + R-D < 0,
That is, d > R + R,
Then, the two circles are separated
Therefore, a
If the diameter of the circle O is 2, the distance from the center of the circle O to the straight line L is m. if there is no real root in the quadratic equation MX? - 2 of X, there is no real root, then the position relationship between circle O and line L is () A. Intersect B. separate C. tangent D. intersect or tangent
（-2√2）²-4m×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
According to the meaning of the question, if the equation has no real root, we can get (R + R) 2-d2 < 0,
Then: (R + R + D) (R + R-D) < 0,
Because R + R + d > 0, R + R-D < 0,
That is, d > R + R,
Then, the two circles are separated
Therefore, a
Given that the radius of the circle O is r, the distance between the line AB and the center O of the circle is D, and the equation x? - 2 √ DX + r = 0 has real roots, then the position relationship between the line AB and the circle O is?
If there is a real root, B ^ 2-4ac > = 0
The substitution is 4d-4r > = 0
d>=R
That is, the relationship between a line and a circle is tangent or apart
Given that the absolute value of OA vector = 1, the absolute value of OB vector = root 3, OA dot multiplication ob = 0, point C is in the angle AOB, and the angle AOC = 30 °, let OC vector = MOA + nob to find M / n The following answer is very difficult to understand, can you explain it in your own words
Answer: M / N = 3 / x0d please refer to the figure below for details
Let the convex quadrilateral ABCD circumscribed to the circle O, and ab = 2, BC = 3, CD = 4, then OA * OC + ob * od = (6 * radical 2), why? Ao = root 3, OC = root 6, OA * OC + ob * od = 2 * Radix 3 * Radix 6 = 6 * Radix 2 How did this come about?
Since the sum of the opposite sides of a circle circumscribed quadrilateral is equal, it is easy to know that Da = 3
So the quadrilateral ABCD is an isosceles quadrilateral, its height is the diameter of circle O, so it is easy to calculate the radius of circle O as the root 2. Then the length of Ao, OC and other segments can be easily calculated
The eccentricity of the ellipse is root 6 / 3, and the distance from one end of the minor axis to the right focus is root 3 The eccentricity of ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is root 6 / 3, the distance from one end point of minor axis to right focus is root 3, (1) solve elliptic equation (2) let line L and ellipse C intersect at two points AB, the distance from coordinate origin o to straight line L is root 3 / 2, and the maximum area of triangle AOB is obtained
(1) the distance from one end of the minor axis to the right focus is root 3, that is, a = 3
From E = C / a = root 6 / 3, C = root 6
a^2=9,c^2=6,b^2=a^2-b^2=3,
So the equation is x ^ 2 / 9 + y ^ 2 / 3 = 1
It is known that the eccentricity e of the ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is known to be the root of 3, and the distance from one end point of the minor axis to the right focus is root 3 Find (1): the equation of elliptic C (2) : let the line L and ellipse C intersect at two points a and B, and the distance from coordinate origin o to line L is the root of 2 / 3, and the maximum area of triangle ABC is obtained
(1）：
e²=c²/a²=2/c
b²+c²=3
b²+a²-b²=3
a²=3,b²=1
The elliptic equation is x 2 / 3 + y 2 = 1
Given that the ellipse C: x square / a square + y square / b square = 1, the eccentricity is 5 / 3, and the distance from one end point of the minor axis to the right focus is 3, the equation of ellipse C is obtained,
One end point of minor axis B (0, b) right focus F2 (C, 0)
|BF2|=√(b^2+c^2)=a=3
e=c/a=√5/3 c=√5
b^2=a^2-c^2=4
The equation of elliptic C
x^2/9+y^2/4=1
It is known that the eccentricity of the ellipse C: x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is 2 / 2 root 3, and the distance from one end of the minor axis to the right focus is 2 If P is a moving point on the ellipse, F1 and F2 are the left and right focal points of the ellipse respectively. Find the maximum and minimum values of vector Pf1 * vector PF2
The distance from one end of the minor axis to the right focus is a a = 2 C = √ 3 / 2, B = 1 ellipse x ^ 2 / 4 + y ^ 2 = 1 F1 (- √ 3,0) F2 (√ 3,0) P (m, n) vector Pf1 = (- M - √ 3, - n) vector Pf1 = (- M + √ 3, - n) vector Pf1 * vector PF2 = m ^ 2 + n ^ 2-3m ^ 2 / 4
RELATED INFORMATIONS
- 1. 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
- 2. 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!
- 3. 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~
- 4. If the radius of ⊙ o is 5cm, and the shortest chord length passing through a point P in ⊙ o is 8cm, then Op=______ .
- 5. What is the chord length of the straight line y = radical 3x cut by the circle x? + y? - 4Y = 0?
- 6. 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______ .
- 7. It is known that the radii of the intersecting circles are 5cm and 4cm respectively, and the common chord length is 6cm
- 8. 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
- 9. 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
- 10. 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
- 11. 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
- 12. 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______ .
- 13. 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
- 14. 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
- 15. 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!
- 16. 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______ .
- 17. 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
- 18. 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
- 19. 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
- 20. 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