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In the plane rectangular coordinate system, if the center of circle P (2, a) (a > 2), radius is 2, and the length of chord ab of circle P is two 3, then the value of a is I know the answer is 2 + radical 2, but why
Two ideas:
1. The equation of circle is listed, and then, the solution of the system of equations composed of two equations, with a, can be found. Then the distance between the two intersection points can be calculated, and the value of a can be obtained by substituting the chord length value
The circular equation is (X-2) 2 + (Y-A) 2 = 4, which is a simultaneous system of equations with y = X
2. You can draw a picture and look at the triangle formed by two points and the center of the circle. Then, according to the angle relationship, you can know what the angle of the center of the circle is, and then you can solve the problem according to the relationship between the triangle
As shown in the figure, in the plane rectangular coordinate system, the circle center of ⊙ P is (2, a) (a > 2), the radius is 2, and the length of chord AB cut by ⊙ P for the image of function y = x is 2 3, then the value of a is () A. 2 Two B. 2+ Two C. 2 Three D. 2+ Three
Through point P, make PE ⊥ AB to e, pass through point P to make PC ⊥ X axis to C, cross AB to D, and connect PA
∵PE⊥AB，AB=2
3, radius 2,
∴AE=1
2AB=
3，PA=2，
According to Pythagorean theorem: PE=
22−(
3)2=1，
∵ point a is on the line y = X,
∴∠AOC=45°，
∵∠DCO=90°，
∴∠ODC=45°，
The △ OCD is an isosceles right triangle,
∴OC=CD=2，
∴∠PDE=∠ODC=45°，
∴∠DPE=∠PDE=45°，
∴DE=PE=1，
∴PD=
2．
The center of ⊙ P is (2, a),
∴a=PD+DC=2+
2．
Therefore, B
As shown in the figure, in the plane rectangular coordinate system, take a (3,0) as the center of the circle, and cut the chord BC = 8 by the Y axis of the circle a, and find the radius of the circle a
The radius is the root (8 △ 2) ^ 2 + 3 ^ 2 = root 25 = 5
In the plane rectangular coordinate system, the center of circle a is on the X axis, the radius is 1, and the analytic formula of the straight line L is y = 2x-2. If circle a moves right along the X axis, when there is a common point between circle a and l, what is the maximum distance of point a moving?
The title is incomplete, and the coordinates of the initial center of the circle are not given
Suppose the initial center coordinate is (XO, 0)
Suppose that the equation of the circle after moving is (x-a) ^ 2 + y ^ 2 = 1, and the linear equation is simultaneous. By substituting y = 2x-2 into the circular equation, a quadratic equation of one variable is obtained. Let △ of the equation = 0, there are two solutions to a, and the largest one is taken. Therefore, the center of the circle after translation is (a, 0). Therefore, the maximum distance of a moving is a-xo
In the plane rectangular coordinate system, the line y = - 2x + 12 intersects point a with X axis, B with y axis and y = x with line y = x at point C. if point P is the moving point on line OA (excluding two points a / O), the line OC crosses point D and connects PC. let OP = t, the area of triangle PDC is s, and the direct functional relationship between S and t is obtained
According to y = - 2x + 12 and y = x, point C (4,4), point a (6,0), CE = 4, so the area of triangle COA is (OA * ce) / 2 = 12, the area of triangle CPA is (AP * ce) / 2 = 12, and because PD is parallel to AB, the area ratio of triangle ODP to triangular OCA is similar
In the plane rectangular coordinate system, a and B are the moving points on the x-axis and y-axis respectively. If the circle C with the diameter of AB is tangent to the straight line 2x + y-4 = 0, the minimum area of circle C is () A. 4 5 pi B. 3 4 pi C. （6-2 5）π D. 5 4 pi
∵ AB is the diameter, ∵ AOB = 90 °,
The o point must be on the circle C,
Make a vertical line from O to the straight line, and the foot of the perpendicular is d. when D is exactly the tangent point between the circle and the straight line, the radius of circle C is the smallest, that is, the area is the smallest
The diameter of the circle is O and the distance from the line is 4
5, then the area of circle C is π× (2)
5）2=4π
5．
Therefore, a
As shown in the figure, in the plane rectangular coordinate system, the straight line AB intersects the x-axis positive half axis at a, and the intersection Y-axis negative half axis at B. (1) if ob / OA = 2 / 3, ab = 2 root sign 3, find the straight line ab
Let Ao = 3x, Bo = 2x, so 9x ^ 2 + 4x ^ 2 = 12, so x = 2 / 13 * root sign 39
So Ao = 6 / 13 times root number 39 Bo = 4 / 13 times root number 39
So a (6 / 13 times root number 39, 0), B (0, - 4 / 13 times root number 39)
After using the two point method, we can find the analytic formula of the line ba
In the plane rectangular coordinate system, the incoming light L1 parallel to the X axis and passing through the point a (3 roots, 3,2) is reflected by the straight line y = 3 root sign 3x / 3, and the reflected light L2 intersects the Y axis of B, In the plane rectangular coordinate system, the incoming light L1 parallel to the X axis and passing through the point a (3 and 3,2) is reflected by the line y = 3 root sign 3x / 3. The reflected ray L2 intersects the Y axis of B, the circle C passes through point a and is tangent to L1 and L2 (1) Find the equation of the line where L2 is located and the equation of circle C (2) Let P and Q be the moving points on line L and circle C respectively, and find the minimum value of Pb + PQ and the coordinates of point P at this time
(1) If the slope is k, and the inclination angle is α, Tan α = ktan (α - β) = (Tan α - Tan β) / (1 + Tan α· Tan β) the incident light K1 = 0, the straight line k = √ 3 / 3, the reflected light K2, the incident angle = the reflection angle (α - α 1 = α 2 - α) has (...)
The image of the first order function can be obtained by translating the line y = 3x, and the area of the triangle formed by it and the line y = - 3x and X axis is six. Find the intercept of the linear function on the Y axis and the area of the triangle formed by it and the coordinate axis
Let y = 3 (x + a) + B, then the intersection point is: 3 (x + a) + B = - 3x. The coordinates of the intersection point are (- (3a + b) / 6, - (3a + b) / 2), and its intersection with X axis is (- (3a + b) / 3,0. Thus, the area of the triangle surrounded by the straight line y = - 3x and X axis is: (3a + b) / 3 * (3a + b) / 2 * 1 /
The area of the triangle formed by the line y = - 3x + 4 and X axis and Y axis is
Let x = 0 and y = 4
Let y = 0, x = 4 / 3
The triangle area of image and coordinate axis besieged city: 1 / 2 × 4 × 4 / 3 = 8 / 3
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