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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
Connect EC, OC, OD, Mo and extend Mo to n
Then the angle e = angle COD / 2 = angle con, so o, m, e, C are in the same circle
Conclusion
As shown in the figure AB is the diameter of circle O and CD is the chord. Through C and D, we make CN vertical CD, DM vertical CD, respectively, and cross AB to n M. are an and BM equal
Because 0 is the center of the circle, CD is two points on the circle. So OC = OD, both are radius. Triangle OCD is isosceles trapezoid. Make the height of CD side, this height must be perpendicular to CD. So it is parallel to MD and NC
Given the fixed point a (- 2,0), the moving point B is a point on the Circle F (X-2) ^ 2 + y ^ 2 = 64 (F is the center of the circle), and the vertical bisector of line AB intersects BF at P.1. Then the trajectory equation of moving point P is 2. The locus of point P intersected by the straight line y = √ 3x + 1 is at M and N. if there is a point C on the trajectory of point P, it is vector om + vector on = m times vector OC, and the value of real number m is?
If the moving point P is an ellipse focusing on a f, then 2A = 8, C = 2, B ^ 2 = a ^ 2-C ^ 2 = 12, then the trajectory equation of the moving point P is x ^ 2 / 16 + y ^ 2 / 12 = 12. Let m (x1, Y1) n (X2, Y2) C (x, y) replace the straight line y = √ 3x + 1 into
The coordinates of point a are known to be (- 1) B is a circle F: (x-1) 2) 2 + y2 = 4, the vertical bisector of segment AB intersects BF at P, then the trajectory of moving point P is () A. Circle B. Ellipse C. A branch of hyperbola D. Parabola
From the meaning of the title, | PA | = | Pb |,
∴|PA|+|PF|=|PB|+|PF|=r=2＞|AF|=1
The locus of point P is an ellipse focusing on a and F
Therefore, B
The coordinates of point a are known to be (- 1) B is a circle F: (x-1) 2) 2 + y2 = 4, the vertical bisector of segment AB intersects BF at P, then the trajectory of moving point P is () A. Circle B. Ellipse C. A branch of hyperbola D. Parabola
From the meaning of the title, | PA | = | Pb |,
∴|PA|+|PF|=|PB|+|PF|=r=2＞|AF|=1
The locus of point P is an ellipse focusing on a and F
Therefore, B
There is a point P (1,1) in the circle (x-1) ^ 2 + y ^ 2 = 4. AB passes through the point P. if the chord length AB = 2 and the root sign 3, then the square of the line
According to the problem, the distance from the center of the circle to the straight line is 1, and the slope of the line exists
Let the slope be K, then the equation is y = K (x-1) + 1 / √ (1 + k?) = 1
Then k = 0 and the equation is y = 1
In a circle of radius 3, there is a chord ab of length 3. Find the length of the arc to which chord AB is opposite What is the length of the arc
Because the radius is 3 and the chord length is 3, then AOB is an equilateral triangle. Then the angle of the center of the circle to which chord AB is opposite is 60 degrees, that is π / 3
Then the length of the arc is the center angle × radius = π
As shown in the figure, in ⊙ o, AB=2 CD, try to judge the relationship between AB and CD, and explain the reason
AB＜2CD．
take
If the midpoint e of AB connects EA and EB, then
EA=
EB=
CD，
So EA = EB = CD,
In △ Abe, AE + be > AB, i.e., 2CD > ab,
Then AB < 2CD,
∴CD＜AB＜2CD．
In a circle with radius 9, there is a chord ab of length 9. What is the length of the arc that chord AB is opposite to
One sixth of the circumference
nine point four two
In a circle O of radius 10, the length of chord AB is 10 (1) Find the size of the central angle α of the circle to which the string AB is opposite; (2) Find the arc length L of the sector where α is located and the area s of the arc
(1) From the radius of ⊙ o = 10 = AB, we know that ⊙ AOB is an equilateral triangle,
∴α=∠AOB=60°=π
3．
(2) According to (1), α = π
3, r = 10, ν arc length L = α· r = π
3×10=10π
3，
/ / s sector = 1
2lr=1
2×10π
3×10=50π
3，
And s △ AOB = 1
2•AB•10
Three
2=1
2×10×10
Three
2=50
Three
2，
ν s = s sector-s △ AOB = 50 (π
3−
Three
2)．
RELATED INFORMATIONS
- 1. 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
- 2. 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
- 3. 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
- 4. 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
- 5. 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]
- 6. In the triangle ABC, ab = ac.f on AC, AE = AF is intercepted on the extension line of Ba to verify EF vertical BC
- 7. If the tangent point of a circle is a tangent point of a circle, then the tangent point of the circle is abo Take yourself in the picture below
- 8. Known, as shown in the figure, ab ‖ CD, AE bisection ∠ BAC, CE bisection ∠ ACD, find the degree of ∠ E
- 9. As shown in the figure BD is the diameter of circle O, ab = AC, ad intersects BC at point E, AE = 2, ed = 4 The first question is to find the length of AB, the second question is to extend DB to f so that BF = Bo, connect FA, and prove that FA is tangent to circle o
- 10. As shown in the figure, ad vertical BC BD = DC point C on the vertical bisector of AE, what is the relationship between the length of AB AC CE and ab + BD and de 1. As shown in the figure, ad vertical BC, BD = DC, point C is on the vertical bisector of AE, what is the relationship between the length of AB AC CE and ab + BD and de?
- 11. 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
- 12. 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
- 13. 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)
- 14. 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
- 15. 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
- 16. 0
- 17. (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
- 18. 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
- 19. 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
- 20. 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