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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
（x+1)²+（y-2）²=1
Drawing the graph, we can get the line y = 1
In addition, there is another formula, which is based on the distance between a point and a straight line, which can be quickly obtained
If the distance from the center of the circle x-y-2x-4y = 0 to the straight line passing through the origin is 1, then the equation of the line is（ As the title
Square of circle x + square of Y - 2x-4y = 0
(x-1)^2+(y-2)^2=5
Center (1,2)
The linear equation y = KX passing through the origin
|k-2|/√(1+k^2)=1
(k-2)^2=1+k^2
-4k+4=1
k=3/4
This linear equation is y = 3x / 4
Given the circle C: x2 + Y2 + 2x + 4Y + 1 = 0, then the linear equation passing through the center of the circle C and having the largest distance from the origin is______ .
The equation of circle C can be changed into (x + 1) 2 + (y + 2) 2 = 4, so the coordinates of the center of the circle are (- 1, - 2) the slope of the line connecting the center of the circle and the origin is − 2 − 0 − 1 − 0 = 2, and the slope of the line with the largest distance from the origin is − 12, and the line passes through the center (- 1, - 2), so the equation is Y - (- 2) = −
It is known that the radius of circle a is 1, the distance between point P and point O is r, and the quadratic equation x 2 - 2x + r = 0
Because the equation has real roots, so △ = 4-4r ≥ 0, R ≤ 1,
The radius of the circle is 1, which means that the distance from point P to the center of the circle is less than or equal to the radius, so point P is in or on the circle
If the circle C with radius r, x ^ 2 + y ^ 2 + DX + ey + F = 0, the distance between the center C of circle C and the straight line L: DX + ey + F = 0 is D, where d ^ 2 + e ^ 2 = f ^ 2 and F > 0 1. Find the range of F 2. Verification: D ^ 2-r ^ 2 is a constant value
(1) (2) substituting the center of circle (- D / 2, - E / 2) into the formula of the distance between the point and the line, d = | - D? 2-e? 2 / 2 + F | / √ (D | + e ﹤)
It is known that the radius of the circle O is 10, the distance from the center of the circle O to the straight line L is od = 6. There are three points a, B and V on the line L, ad = 6, BD = 8, CD = 5 times the root sign 3 Question: what is the position relationship between point a, B and C and circle O? Question: what is the relationship between a, B, C (the number above is wrong, not V is c) and the position of circle O?
∵ OA = 6 times root number, OB = 10, OC = root number 111, OA = 6 times root number 2 is less than R, OB = R, OC is greater than R
⊙ o, B ⊙ o, C ⊙ o
I made it myself. I don't know if it's right!
It is known that the radius of circle O is 10, the distance od from the center of circle O to the straight line a is 6 cm, there are three ABC points on the line a, and there are ad = 10 cm, BD = 8 cm, CD = 6 cm, respectively Point out the position relation of point a, B, C and circle o
V. the radius of ⊙ o is 10 cm, the distance from center O to line L is od = 6 cm, there are three points a, B and C on the line L, and there are ad = 10 cm, BD = 8 cm, CD = 6 cm. The location relations of points a, B, C and ⊙ o are pointed out respectively
[process] let students draw a graph, combine number and shape, according to Pythagorean theorem, we can get OA = cm, OB = 10cm, OC = cm respectively, and then compare OA, ob, OC and radius
[results] point a is outside ⊙ o, point B is on ⊙ o, and point C is in ⊙ o
If the straight line y = (radical 3 / 3) x + radical 2, and the circle (x-radical 3) ^ 2 + (Y-1) ^ 2 = 3 with the center of the circle D intersect at a and B, then the sum of inclination angles of AD and BD is The answer is that four thirds of PI is urgent
Is ad and straight line?
It is known that the radius of circle O is 5cm, and the length of OD from point O to line L is 4cm. If points a, B and C are on line L, and ad = double root 2cm, If BD = double root 3cm, CD = 3cm, then point a is at, point B is at, and point C is at
Inside, outside, round
It is known that circle C passes through points a (1,4) and B (3, - 2), and the distance between the center of circle C and the straight line AB is Find the equation of circle C
Method I: let C (a, b) be the center of the circle, and the radius is r
The midpoint of the easy to see line AB is m (2,1) (2 points)
∵CM⊥AB，kAB=-2-4
3-1=-3
∴kCM=b-1
a-2=1
3 is: 3B = a + 1 (5 points)
And ∵ cm|=
10∴（a-2）2+（b-1）2=10②… (8 points)
We can get
a=-1
B = 0 or
A=5
B=2
That is C (- 1,0) or C (5,2) (10 points)
∴r2=|CA|2=20
So the equation of the circle is: (x + 1) 2 + y2 = 20 or (X-5) 2 + (Y-2) 2 = 20 (12 points)
Method II: ∵ a (1,4), B (3, - 2)
The equation of line AB is: 3x + Y-7 = 0 (2 points)
∵ the midpoint of line AB is m (2,1)
The center of the circle C falls on the perpendicular line of the line AB: x-3y + 1 = 0 (4 points)
Let C (3b-1, b) (5 points)
∴|3(3b-1)+b-7|
32+12=
10… (8 points)
The solution is b = 0 or B = 2
That is C (- 1,0) or C (5,2) (10 points) ν R2 = | Ca | 2 = 20
So the equation of the circle is: (x + 1) 2 + y2 = 20 or (X-5) 2 + (Y-2) 2 = 20 (12 points)
RELATED INFORMATIONS
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- 2. 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
- 3. 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!
- 4. 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~
- 5. If the radius of ⊙ o is 5cm, and the shortest chord length passing through a point P in ⊙ o is 8cm, then Op=______ .
- 6. What is the chord length of the straight line y = radical 3x cut by the circle x? + y? - 4Y = 0?
- 7. 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______ .
- 8. It is known that the radii of the intersecting circles are 5cm and 4cm respectively, and the common chord length is 6cm
- 9. 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
- 10. 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
- 11. 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______ .
- 12. 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
- 13. 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
- 14. 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!
- 15. 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______ .
- 16. 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
- 17. 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
- 18. 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
- 19. 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
- 20. 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