In the rectangular coordinate system, the distances from a moving point m (x, 0) on the X axis to two points P (5,5) and Q (1,2) are MP and MQ respectively. Find the minimum value of MP + MQ and the coordinate of point M

In the rectangular coordinate system, the distances from a moving point m (x, 0) on the X axis to two points P (5,5) and Q (1,2) are MP and MQ respectively. Find the minimum value of MP + MQ and the coordinate of point M

The analytic formula y = 7 / 4 x - 4 / 15, m (15 / 7, 0)

In the rectangular coordinate system xoy, the distances from the moving point m (x, 0) on the x-axis to the fixed points P (5,5) and Q (2,1) are MP and MQ respectively. Then, when MP + MQ takes the minimum value, the abscissa of point m is______ ．

As shown in the figure, make the symmetric point Q 'of Q about x-axis and connect PQ'. According to the properties of the axisymmetric figure, QM = q ′ m, so QM + MP = q ′ m + MP = q ′ P. according to the shortest line segment between two points, M is the point to be solved. Suppose the analytic formula is y = KX + B, ∵ point Q and point Q ′ are symmetric about x-axis, ∵ Q ′ (2, - 1) substituting P (5,5) and Q ′ (2, - 1) into the analytic formula respectively, 5K + B = 52K + B = - When y = 0, x = 52. The coordinate of m point is (52, 0). So the answer is: 52

In the plane rectangular coordinate system, the angle xoy = 90 degrees, points a and B move on the positive half axis of X axis and Y axis respectively, and be bisects the angle aby.be And the bisector C
(1) If the angle OAB = 30 degrees, find the degree of angle ACB
(2) Does the angle ACB change with the movement of points a and B?

45°
No change
∵∠eba=90°-1/2∠oba
∠bac=45°-1/2∠oba
∴∠c=∠abe-∠bac=45°

As shown in the figure, in the plane rectangular coordinate system, the coordinate of point a is (- 3,0), point B is on the y-axis, ab = 5, AP bisects ∠ Bao and intersects the y-axis on P
(1) Find the coordinates of the symmetric point B 'of point B about the straight line AP;
(2) If M and N are the moving points on the positive half axis of AB and X axis respectively, and PM = PN is maintained, Q: does the value of am + an change in this process? If it does not change, calculate its value; if it changes, calculate its range of change

Let p be PC perpendicular to C, and according to AP bisection, the Y-axis of Bao intersects with P. it can be proved that PAC and Pao are congruent (bisection angle, etc., vertical angle, etc., collinear). From this, AC = Ao. So AB and X-axis are symmetric about AP. Because B and B 'are symmetric about AP, B'x-axis is. Because AB = 5, ab'