Knowing that the two points A 0,2 , B 4,1 , and the point P are the points on the x axis, find the m

Point A: The point of symmetry with respect to the x-axis is A'(0,-2), connect a'b According to the shortest straight line between two points, the minimum value of Pa+PB is |a'b|, where point p is a'The intersection of b and x-axis|a'b|= [4 2+(1-(-2)) 2]=5, so the minimum value is 5

Take point A with respect to the x-axis symmetry point A (0,-2), and pass points A and B as straight lines L

The straight line L intersects the x-axis at point p.

Let the equation for the line l be y=kx+bCrossing the points a (0,-2) and b(4,1) gives k=3 4,b=-2The equation for the straight line l is y=3 4 x-2 when y=0 and x=8 3So the p-point coordinates (8 3, 0).

Do point a(0,2) with respect to the x-axis and the point of symmetry is a'(0,-2) straight line a'The intersection point of b and the x-axis is the minimum point p of the pa+pb line, a'The equation for b: y-1=3(x-4) 4y=0, x=8 3 point p coordinates (8 3,0).

point a with respect to the x-axis symmetry of a (0-2), connect a b, and the intersection point with the x-axis is p, find the distance of a b [root number (0 4) 0 5 + ((2) 1) 0 5 = 5

Find the point of symmetry a with respect to the x-axis'Connect A'b, the intersection point with the x-axis.

<> this limb as shown in Fig., do b with respect to the x-axis symmetry point b1 (8-2), connect ab1, the intersection of the x-axis at the p point, then there is a minimum value of pa+pb at this time, pass b1 to make the y-axis perpendicular, let the point be c, in the right-angled source triangle acb1, the two right-angled edges are respectively, then according to the pythagorean theorem, the hypotenuse is 10, or according to the distance between the two points formula, the first state ab1=pa+pb=10 can be obtained

Start by finding the point a, the point of symmetry a, with respect to the x-axis'(0, 2) and then straight a'The intersection of b and x is the p point.

Straight line a'b:(y+2) x=(1+2) 4 let y=0.

x=8 3 so p(0,8 3).

Point A: The point of symmetry with respect to the x-axis is A'(0,-2), connect a'b According to the shortest straight line between two points, the minimum value of Pa+PB is |a'b|, point p is.

a'The intersection of b and x-axis|a'b|= [4 2+(1-(-2)) 2]=5, so the minimum value is 5

Point A: The point of symmetry with respect to the x-axis is A'(0,-2), connect a'b According to the shortest straight line between two points, the minimum value of Pa+PB is |a'b|, point p is.

a'The intersection of b and x-axis|a'b|= [4 2+(1-(-2)) 2]=5, so the minimum value is 5

Dear landlord:

point a with respect to the x-axis symmetry of a (0-2), connect a b, the intersection point with the x-axis is p, find the distance of a b [root number (0 4) +2) 1) =5 I wish you a high step.

Looking forward to your adoption, thank you.

As point B, the point C is symmetrical with respect to the X-axis, and the intersection point of AC and the X-axis is the point P that satisfies the condition

The sum of the two sides of the triangle is greater than the third side.

Since C is symmetrical with B with respect to the X axis, P is a point on the X axis, then PB=PC, if P is not on AC, then in APC, there is always PA+PC AC, and only when P is on AC, there is PA+PC=AC, so PA+PC AC, i.e., PA+PB AC].

b(4,1)，c(4，-1)

Let the analytic formula of the straight line where ac is located be y=kx+b, and substitute a(0,2) and c(4,-1) into :

b=2，4k+b=-1

k=-3/4，b=2

That is, the analytical formula of the straight line where ac is located is y=-3 4x+2

Let y=0, then x=8 3

That is, the p-point coordinates of the full envy and acceptance of the conditions are the eggplant bridge (8 3, 0).

Do point a with respect to the x-axis of the symmetry point a', then the a' coordinate is (0,2).

Connect A'B intersects the x-axis at a point, which is the point p, and Pa+PB is the smallest.

Pa+PB=[(4-0)2+(-1-2)2]=5 under the root number

Let p(x,0), the square of x + the square of 4 + the square of (x-8) + the square of 2 = the square of 2*x-16*x+84 find the minimum value, and the square of =2*(x-4) + 52, so x=4

Start by drawing a planar Cartesian coordinate system.

Indicate points A and B.

Make point a with respect to the x-axis of the symmetry point a'

Connect A'B is the line segment BC perpendicular to the Y axis.

At this time, the length of PA plus PB becomes PA'and PB when A'When bp three points are on the same line. There is a minimum value of Pa'+Pb, which is the minimum value of Pa+Pb.

a'The length of c is the ordinate of point b plus a''s ordinate.

The length of BC is the abscissa of point B.

The Pythagorean theorem.

The root number gets A'B length 10

So the minimum value of Pa'+Pb is 10, that is, the minimum value of Pa+Pb is 10

Connect AB and extend AB to the X axis at a point, which is the P point.

In a triangular PAB, PA-PB is the maximum: root number (8 2 + (4-2) 2) = root number 68 = 2 root number 17

pa-pb < = ab = root number [(-8) 2 +2 2] = 2 root number (17).

When a, b, p are collinear, the minimum value is 2 root numbers (17).ans

At this point p(p,0) => (4-2) (0-2)=(0-4) (p-0) =>-1=-4 p => p=4 =>p(4,0).

The point of symmetry of solution a with respect to the x-axis is a'(0,-4).

pa+pbpa’+pb

a'b|=10

a, with respect to the x-axis, the symmetry point is c(0,-4).

then Pa=PB

BC on either side of the x-axis.

So P is the intersection of the straight line BC and the X axis.

bc is x+4y-16=0

y=0，x=16

So p(16,0).

Do b with respect to the x-axis symmetry point b'(8,-2) and then connect ab'And it came out.