Given the circle x ^ 2 + y ^ 2 + 2x-4y-1 = 0 and a point m (4, - 1) outside the circle, make the tangent of the circle through M, and the tangent points are D and E. calculate the length and length of the tangent

Given the circle x ^ 2 + y ^ 2 + 2x-4y-1 = 0 and a point m (4, - 1) outside the circle, make the tangent of the circle through M, and the tangent points are D and E. calculate the length and length of the tangent

Let the center of the circle be o, the tangent length ^ 2 = om ^ 2-r ^ 2 = 34-6 = 28, that is, the tangent length is the root sign 28
Ed vertical OM, in right triangle OMD, OM * (0.5de) = R * tangent length, de can be obtained

Let a point P (x0, Y0) outside the circle (X-2) ^ 2 + (Y-3) ^ 2 = 1 be tangent to the circle, and the tangent point M.O is the origin. | PM | = | Po |

Let C (2,3), then
|PM|^2=|PO|^2=|PC|^2-1
That is: x0 ^ 2 + Y0 ^ 2 = (x0-2) ^ 2 + (y0-3) ＾ 2-1
The result is: 2x0 + 3y0-6 = 0
When PM is the smallest, Po is the smallest
Since the point P (x0, Y0) satisfies 2x0 + 3y0-6 = 0, the trajectory of point P is a straight line 2x + 3y-6 = 0
When Po is minimum, the minimum value is the distance from point O to line 2x + 3y-6 = 0
The equation of a line passing through point O and perpendicular to the line 2x + 3y-6 = 0 is y = (3 / 2) X
The simultaneous equations 2x + 3y-6 = 0 and y = (3 / 2) x are used to obtain the coordinates of point P such that | PM | is minimum: (12 / 13,18 / 13)

Given that the equation of a circle is x ^ 2 + y ^ 2 + 2x-8y + 8 = 0, a tangent of the circle is made through P (2,0), and the tangent point is a, then the length of PA is

Root (5 ^ 2-3 ^ 2) = 4

It is known that the equation of a circle is X & # 178; + Y & # 178; + 2x-8y + 8 = 0. If the tangent point is a, then the length of PA is?
Please answer the teacher, please write the detailed steps. Thank you!

The circle equation can be reduced to (x + 1) &# 178; + (y-4) &# 178; = 9, so the center O of the circle is (- 1,4), and the radius is 3. Drawing PA & # 178; + R & # 178; = L & # 178;
L is the distance between P and O, l = 5, r = 3, so PA = 4