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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!
The circle equation is (x + 1) ^ 2 + (y + 2) ^ 2 = 8
It is easy to know that the center of the circle is (- 1, - 2), and the radius is 2 √ 2
Drawing of circles and lines in the same coordinate system
It is obvious that the position relationship between the line and the circle is intersection
It is easy to know that the set of points whose distance to the known straight line is a certain value is two lines parallel to the line
Then the intersection of the two lines parallel to the known straight line and the circle is the point of intersection
It should be noted that the relationship between the two lines and the circle may be intersection, tangency or separation
Let the equation of parallel line: x + y + M = 0
Obviously, the distance between the parallel line and the straight line x + y + 1 = 0 is √ 2
According to the distance formula between parallel lines, there is | 1-m | / √ (1 ^ 2 + 1 ^ 2) = √ 2
M = - 1 or M = 3
Therefore, the equations of two parallel lines are x + Y-1 = 0 and X + y + 3 = 0
The above two linear equations are combined with the circular equation respectively
That is (x, y) = (1,0) by solving the equations (x + 1) ^ 2 + (y + 2) ^ 2 = 8 and X + Y-1 = 0
Indicates that the line is tangent to the circle
By solving the equations (x + 1) ^ 2 + (y + 2) ^ 2 = 8 and X + y + 3 = 0, we get (x, y) = (- 3,0) or (1, - 4)
Indicates that a line intersects a circle
It can be seen that there are three points satisfying the condition, which are (1,0), (- 3,0) or (1, - 4)
If we can make a more accurate graph, it is not difficult to find that the distance between the center of the circle (- 1, - 2) and the straight line x + y + 1 = 0 is exactly √ 2, and the radius of the circle is twice √ 2, Three points can be determined by simple geometric method
If there are at least three different points on the square of circle x plus the square of Y minus 4x minus 4Y minus 10 and the distance from the line y = KX is 2 times the root sign, then the value of K is taken
(x-2)^2+(y-2)^2=18
If there are at least three different points, the distance from the line y = KX is twice as long as the root 2, that is, the line to the center of the circle is root 2 (drawing, two lines intersect and one tangent)
Draw a circle and a straight line. The angle from the origin to the center of the circle is 2 root sign 2, so the angle is 30 ° which means tan15 to tan75, 2-radical 3 to 2 + radical 3
According to the equation | 2-2k | = radical (2 (k ^ 2 + 1)), the result is also a closed interval from 2-radical 3 to 2 + radical 3
The distance from the circle x2 + Y2 + 2x + 4y-3 = 0 to the straight line x + y + 1 = 0 is Points of 2 share () A. 1 B. 2 C. Three D. Four
By transforming the equation of circle into standard equation, we get: (x + 1) 2 + (y + 2) 2 = 8,
The coordinates of the center of the circle are (- 1, - 2), and the radius is 2
2，
The distance from the center of a circle to the straight line x + y + 1 = 0 d = 2
2=
2，
Then the distance from the circle to the straight line x + y + 1 = 0 is
There are three points of 2
Therefore, C is selected
The radius of ⊙ o is known to be 10cm. If the distance between the center O and a straight line is 10cm, then the position relationship between the line and the circle is () A. Separation B. Tangency C. Intersection D. Can't be sure
⊙ the radius of O is 10 cm, and the distance from the center O to a straight line is 10 cm,
The straight line is tangent to the circle
Therefore, B
The distance from the center of the circle to the center of the circle is 10 cm
Intersecting relationship
When two circles with equal radii intersect, the distance between the centers of the two circles is equal to the radius and the radius is equal to 10 cm
Because the distance between the centers of a circle is the radius, so the center of a circle connects two intersections on the other circle, then the shadow becomes two arches. For one of them, the area of the arch is fan-shaped triangle, so we need to connect the radius
Given that the diameter of the circle is 13cm and the distance from the center of the circle to the straight line L is 6cm, then the number of common points between the line L and the circle is______ .
The radius of the circle is 6.5cm
Because the distance between the center of the circle and the line L is 6cm, the line and the circle intersect, so there are two intersections
(2012 · Hengyang) it is known that the diameter of ⊙ o is 12cm, and the distance between the center of circle O and the line L is 5cm, then the number of intersections between line L and ⊙ o is () A. 0 B. 1 C. 2 D. Can't be sure
According to the meaning of the title, we get
If the radius of the circle is 6cm, that is, 5cm larger than the distance from the center of the circle to the straight line, the straight line intersects the circle,
Therefore, the number of intersection points between line L and ⊙ o is 2
Therefore, C
It is proved that the distance between the tangent line and the center of the circle is equal to the radius of the circle
It is divided into three steps
(1) Suppose that the tangent at is not perpendicular to the radius OA of the tangent point,
(2) At the same time, make a vertical line om of at. Through proof, we get the contradiction. If om ⊙ OA is the radius, then there is a quantitative relationship in the position relationship between a straight line and a circle
(3) At ⊥ Ao
If the distance from a point to the center of a circle is equal to the radius, then the point is outside the circle If () is less than the radius, the point is in the circle A circle can be determined by passing through three points of () The circle passing through the three vertices of a triangle is called triangle (), and the center of the circle is called () The outer center of a triangle is the intersection of () and its distance to () is equal
If the distance between a point and the center of a circle is greater than the radius, then the point is outside the circle. If the distance from the point to the center of the circle is equal to the radius, then the point is in the circle. If the distance from the point to the center of the circle is less than the radius, then the point is in the circle
RELATED INFORMATIONS
- 1. 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~
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- 3. What is the chord length of the straight line y = radical 3x cut by the circle x? + y? - 4Y = 0?
- 4. 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______ .
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- 11. 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
- 12. (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
- 13. 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
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- 15. 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
- 16. 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
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- 18. 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______ .
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