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If the center of the circle is (3b, b), then the radius is 3|b|
The distance from the center of the circle to y=x d=|3b-b|/ V 2= V 2|b|
∵(√7)²+(√2|b|)²=(3|b|)²
It is found that, B = ± 1
The equation for a circle is: (x-3) 2 + (Y-1) 2 = 9, or (x + 3) 2 + (y + 1) 2 = 9

Find tangent to the x-axis, the center of the circle C is on the line 3x-y = 0, and the chord length of the cut line X-Y = 0 is 2 The equation for a circle of 7

If the center of a circle (T, 3T) is tangent to the x-axis, the radius r = 3|t |
∵ distance from center of circle to straight line d = | t − 3T|
2=
2t，
ν from R2 = D2+（
7) 2, t = ± 1
The center of the circle is (1, 3) or (- 1, - 3), and the radius is equal to 3
The equation of circle C is (x + 1) 2 + (y + 3) 2 = 9 or (x-1) 2 + (Y-3) 2 = 9

A circle is tangent to the y-axis, its center is on the straight line x-3y = O, and the chord length cut on the line y = x is 2 root sign 7. Find the equation of the circle

Let the center of a circle (3T, t), then the radius is | 3T |, and make a vertical line from the center of the circle to y = X. the distance formula from the point to the straight line and the Pythagorean theorem are obtained
|The result is: (x-3) ^ 2 + (Y-1) ^ 2 = 9
Or (x + 3) ^ 2 + (y + 1) ^ 2 = 9

From the point (4,0) to the circle x square + y square arbitrary secant line, intersect the circle at a and B two points, the trajectory equation of midpoint P of chord AB is obtained X square + y square = 4

Let P (x, y) be the midpoint P (x, y) of a (x1, Y1) B (X2, Y2) point P (x, y), then there will be X1 + x2 = 2x, Y1 + y2 = 2Y, then X1 ^ 2 + Y1 ^ 2 = 4 x2 ^ 2 + Y2 ^ 2 = 4, subtract (x1 + x2) (x1-x2) + (Y1 + Y2) (y1-y2) (y1-y2) = 0, that is, k = (y1-y2) / (x1-x2) = - X / Y and y = K (x-4) eliminate K to get x ^ 2 + y ^ 2-4x = 0 (x ∈ [0,0,0,0,0,0,x ∈ [0,0,0,0,0,0,x ∈ [0,0,0 1]) is the trajectory equation of the middle point

From the point P (5,12) outside the circle x square + y square = 9, the straight line intersects the circle a and B, and the trajectory equation of midpoint m of chord AB is obtained

Therefore, x ^ 2 + y ^ 2 = 9x ^ 2 + [K (X-5) + 9] can be obtained: x ^ 2 + K ^ 2 * x ^ 2-10k ^ 2 * x + 24kx + 25K ^ 2 + 144-9 = 0 (k ^ 2 + 1) x ^ 2 - (10K ^ 2-24k)

It is known that there is a point P (- 1,2) in the square of circle x + the square of y = 8. AB is the chord passing through the point P. the trajectory equation of the midpoint of the chord passing through the point P is obtained

If the midpoint of AB is m, then OM is perpendicular to AB, that is, the angle OMP is 90 degrees,
Therefore, the trajectory of M is a circle with the diameter of OP, i.e. (x + 1 / 2) ^ 2 + (Y-1) ^ 2 = 5 / 4

Over ellipse x2 9+y2 If a certain point (1,0) in 4 = 1 is a chord, then the trajectory equation of the midpoint of the chord is______ .

Let the coordinates of the two ends of the chord be (x1, Y1), (x2. Y2), and the coordinates of the midpoint of the chords are (x, y). The slope of the line where the chord is located is kx219 + y214 = 1x229 + y224 = 1. By subtracting the two formulas, 19 (x1 + x2) (x1-x2) + 14 (Y1 + Y22) (y1-y2) = 0, that is, 2x9 + 2y4k = 0 and ∵ k = YX − 1

Try to determine the trajectory equation of the midpoint of the chord of ellipse x ^ 2 + y ^ 2 / 4 = 1 through the point m (0,1)

The midpoint of the chord ab of M is pa (x1, Y1), B (X2, Y2), P (x, y) then x = (x1 + x2) / 2, y = (Y1 + Y2) / 2 (1) and (y2-y1) / (x2-x1) = (Y-1) / X (2) take AB coordinate into elliptic equation, get X1 ^ 2 + Y1 ^ 2 / 4 = 1x2 ^ 2 ^ 2 + Y2 ^ 2 + Y2 ^ 2 / 4 = 1, two formula subtractiget (x2 + x1) (x2-x1) + (Y2 + Y1) (Y2 + Y1) (y2-y1) (y2-y1) / 4 = 0 (0) put (1) the (x2 + x1) (x2-x1) (x2 + Y1) (Y2 + Y1) (Y2 + Y1) (y2-y1) (y2-y11

Through a certain point (1,0) in the ellipse x ^ 9 + y ^ 4 = 1 as a chord, the trajectory equation of the midpoint of the chord is obtained

Let K (x-1) = y, the midpoint is (x0, Y0) and the midpoint is on the chord, so K (x0-1) = Y0 In
Put the equation of string into the elliptic equation, get: (9K ^ 2 + 4) x ^ 2-18k ^ 2-18k ^ 2x + 9K ^ 2-36 = 0, set the two ends of the string is (x1, Y1) (X2, Y2), and great theorem get X1 + x2 = 18K ^ 2 / (9K ^ 2 + 4), so x0 = (x1 + x2) / 2 = 9K ^ 2 / (9K ^ 2 + 4), so x0 = (x1 + x2) / 2 = 9K ^ 2 / (9K ^ 2 + 4), Y1 + y2 = K (x1-1) + k (x2-1) = K (x1 + x2) - 2K = - 8K / (9K ^ 2 + 4), so Y0, Y0, so Y0, so Y0, so Y0, so Y0, so Y0, so Y0 = 9 K (so x0 / Y0 = - 9 / 4K, so k = - 4x0 / 9y0 can be obtained by substituting the equation into one

Over ellipse x2 9+y2 If a point m (2,0) in 4 = 1 introduces the moving chord ab of an ellipse, then the trajectory equation of midpoint n of string AB is___ .

Let n (x, y), a (x1, Y1), B (X2, Y2), then x129 + y124 = 1 ①, x229 + y224 = 1, ② ① - ②, we can get: (x1-x2) x9 + (y1-y2) Y4 = 0 ? y1-y2x1-x2 = - 4x9y ∵ the moving chord AB passes through the point m (2, 0), and the midpoint of chord AB is n. when m and n do not coincide, there is a K AB = yx-2  yx-2 = - 4x9y