When ma * MB is the minimum value, the equation of line L is obtained

When ma * MB is the minimum value, the equation of line L is obtained

First of all, let's assume that the equation of this line is y = KX + B, and note that K here must be negative
Because point m is one of them
Then B = 1-2k
The coordinates of the intersection point with the X axis are (1-2k, 0)
The coordinate of the intersection point with y axis is (0,2k-1 / k)
Then the length of Ma and MB can be found in the right triangle
Square of Ma = 4K ^ 2 + 4
Square of MB = 1 + 1 / K ^ 2
So ma * MB = - 2 * (k ^ 2 + 1) / k = (2k + 2) / (- K) = - 2 (K + 1 / k)
Then the minimum value of Mamb, that is, when k is a number - (K + 1 / k) is the minimum, that is, K + 1 / K is the maximum
It is not difficult to know that K + 1 / K is less than or equal to - 2
K=-1
So the equation of this line is y = - x + 3

When passing through point P (1,4), make the intersection of line L, x-axis, y-axis, positive half axis at two points a and B, O is the coordinate origin, when OA + ob takes the minimum value, solve the linear l equation
To solve the problem in detail

Let the slope of the straight line be K, because the straight line intersects the positive half axis of X axis and Y axis respectively, so K0 when y = 0, x = |oa | = (K-4) / k > 0 |oa | + |ob | = (4-K) + (K-4) / k = 4-K + 1-4 / k = (- K) + (- 4 / k) + 5. Because - k > 0, - 4 / k > 0, so - K + (- 4 / k) > = 2 radical (- k * (- 4 / k)) = 4. Then the minimum value = 5 + 4 = 9 when - k = - 4 / K

Make a straight line L through P (3,1) and the positive half axis of X, Y axis intersect at two points a and B, and O is the origin. When PA Pb takes the minimum value, find the equation of the straight line L
Please use the method of X / A + Y / b = 1. The process should be as detailed as possible, thank you

Let AB: X / A + Y / b = 1, it passes through P (3,1),
∴3/a+1/b=1,1/b=(a-3)/a,b=a/(a-3),
A(a,0),B(0,b),
Let w = | PA | Pb |, then
w^2=[(a-3)^2+1][9+(b-1)^2]=[(a-3)^2+1][9+9/(a-3)^2]
=9[(a-3)+1/(a-3)]^2=9[|a-3|+1/|a-3|]^2>=36,
When | A-3 | = 1, a > 3, i.e. a = 4, take the equal sign, then B = 4,
The equation of line L is x / 4 + Y / 4 = 1, that is, x + y-4 = 0