The intercept of the straight line of the point p 1,4 is positive, and the sum of the intercepts is

Let the intercept on the x-axis be a and the intercept on the y-axis be b

Then the equation for the straight line can be written from the intercept formula: x a+y b=1 because the straight line passes through the point p(1,4), so 1 a+4 b=1,a>0,b>0a+b (a+b)(1 a+4 b).

1+(b/a)+(4a/b)+4

5+[(b/a)+(4a/b)]

5+2 [(b a)(4a b)] The minimum value of the mean inequality i.e. a+b is 9

When b a=4a b, b=2a, a=3, b=6, take the minimum value ps: the minimum value of a+b is conditionally limited by the inequality a+b 2 (ab), that is, ab must be a fixed value. However, the question only gives the condition of 1 a+4 b=1, and does not give the condition of ab=fixed value, so a+b 2 (ab) is not available.

Let the intersection point on the x-axis be (a,0).The intersection of y is (0,b) and since the three points are directly on a line, a relation to a b can be obtained.

b=4-4/(1-a)

So a+b=a+4 (a-1)+4

The key here is the handling of A+4 (A-1).

There is a very common inequality: a 2 + b 2> = 2ab (proof: a 2 + b 2-2 ab = (a-b) 2> = 0).

The above equation is equal if and only if a=b.

So it's handled like this.

a+b=a-1+4/(a-1)+5>=2*√[a-1)*4/(a-1)]+5

If and only if a-1=4 (a-1) takes the equal sign, the minimum.

What you said is that when you use a=b, a+b has a minimum value, this is a wrong view.

ps: I only saw the first floor after I finished it. Alas.

Let the slope be k k<0

y=k(x-1)+4

x=0 y=4-k

y=0 x=1-4/k

The intercept and the mean of 4-k + 1-4 k = 5 + (-k - 4 k) > = 9 are not equal.

At this point, k = -2

The straight line is y-2x+6

There are many methods, and the better one is:

Let the intercepts on the x, y, and y axes of this line be a, b, a>0, and b>0 respectively, then the equation for the straight line is: x a+y b=1 (intercept formula), because the straight line passes through the point p(1,4), then 1 a+4 b=1, so a+b=(a+b)*(1 a+4 b)=5+(b a+4a b)>=5+4=9, if and only if b a=4a b, i.e., b=2a=6, take the equal sign.

So the equation for the straight line is: x 3 + y 6 = 1, that is, 2x + y - 6 = 0

It can also be done with the point oblique formula of the linear equation.

Let the equation be x a+y b=1

If the point (1,4) is known to pass through a straight line, then 1 a+4 b = 11 a = 1-4 b

a=b/(b-4)

a+b=(b^2-3b)/(b-4)

Another f(b) = (b 2-3b) (b-4).

Derivative of f(b).

When the derivative is 0.

b=2 or b=6

Since a=b(b-4)>0

So b>4

Considering monotonicity, f(b) takes the minimum value at 6.

So b = 6, a = 3

That is, the equation for a straight line is x 3 + y 6 = 1

(1) Over the origin, at this time it is easy to get the equation is 4x+3y=1, (2) but the origin, set to x a+y a=1, substitute (3,-4) to get a=-1, so at this time the equation x+y+1=0, and then to sum up, you can, I hope to adopt, thank you.

If the intercepts are equal, then the slope is 1, so the equation is y=-x+b, and the points (3, 4) are substituted into 4 3+b, b -1

Let the straight line through p be y=k(x-1)+4, then the intercepts of the straight line on the y,x axis are respectively 4-k, and 1-4 k are both greater than 0, so k<4, and k<0, or k>4, so k<0, and the intercept sum is 4-k+1-4 k, which is greater than or equal to 5+2 times the root sign (-k)(-4 k)=9 equal sign, and if and only if -k=-4 k, i.e., k=-2, is true, so the straight line is y=-2(x-1)+4, i.e., y=-2x+6

a+b)=(a+b)(1 a+4 b)greater than or equal to 2 (4a b*b a)=4, here let the straight line be x a+y b=1, the intercept of the straight line on the two coordinate axes x,y is a,b respectively, and the straight line passes (1,4) so 1 a+4 b=1, so a+b=(a+b)*1=(a+b)(1 a+4 b)=1+4a b+4+b a is greater than or equal to 5+2 times the root number (4a b*b a)=5+4=9, The equal sign is true if and only if 4a b=b a, and both a and b are greater than 0, ·· Here we use the inequality property a 2 + b 2 greater than or equal to 2 times the root number ab, as well as some transformation techniques).

y-4=k(x-1)

x=0,y=4-k>0,k<4

y=0,x=1-4 k>0,k<0 or k>4, so k<0

4-k)(1-4 k)=8-k-16 k=8-(k+16 k)Because k<0, the product of k+16 k<-4+16 (-4)=-8 intercepts is the minimum 8-(-8)=16

k=-4y=-4x+8

Let the intercepts on the x, y, and y axes of this line be a, b, a>0, and b>0 respectively, then the equation for the straight line is: x a+y b=1 (intercept formula), because the straight line passes through the point p(1,4), then 1 a+4 b=1, so a+b=(a+b)*(1 a+4 b)=5+(b a+4a b)>=5+4=9, if and only if b a=4a b, i.e., b=2a=6, take the equal sign.

So the equation for the straight line is: x 3 + y 6 = 1, that is, 2x + y - 6 = 0

Let y-4=k(x-1) be taken by the title, when k 0x=0, y=-k+4y=0, x=-4 k+1-k+4+(-4 k+1)=5+(-k-4 k) 5-2 ((k) (4 k))=5-4=1, if and only if -k=-4 k, i.e., k=-2, the equation l is: y-4=-2(x-1).

Let the line ax+b=y then the intersection point of a+b=4 and the axis (x=0, y=b>0); (x=-b a>0) to find the minimum value of b-b a, b

Let the equation for this line be y=kx+b

Because of the point p(1,4).

So the equation can be written as: y=kx+(4-k) if their intercepts on the two axes are positive: then k<0,4-k>0 so k<0

i4-ki+ik/(4-k)i

The minimum is based on the mean theorem: when the two numbers are equal, the minimum value can be obtained:

Solution: You can set the straight line as x a+y b=1 (a>0,b>0) and substitute the coordinates of the point p(1,4) to obtain: 1 a+4 b=1 using the basic inequality, we get: 1=1 a+4 b 2 4 ab So, 1 16 ab

ab 16 (if and only if the equal sign holds if a=2, b=8) So, the linear equation is: x 2 + y 8 = 1

Solution: Let the point p(1,4) pass through and be on two axes.

intercept on . are all straight line equations with positive numbers.

Yes: x a+y b=1, then: 1 a+4 b=1, a>0 , b>0, so:

a+b=(a+b)*(1 a+4 b)=5+(b a+4a b), because: a>0, b>0, so: b a+4a b 2 4=4, so a+b=5+(b a+4a b) 9, when a+b obtains the minimum value of 9, b a=4a b, i.e.:

b=2a, substituting 1 a+4 b=1 obtains: a=3, b=6, therefore: the equation for the straight line is:

x 3 + y 6 = 1, i.e. 2x + y = 6

There are many methods, and the better one is: let the intercepts on the x, y, and y axes of this line be a, b, a>0, b>0, then the linear line equation is: x a+y b=1 (intercept formula), because the straight line passes through the point p(1,4), then 1 a+4 b=1, so a+b=(a+b)*(1 a+4 b)=5+(b a+4a b)>=5+4=9, if and only if b a=4a b, that is, b=2a=6, take the equal sign.

So the equation for the straight line is: x 3 + y 6 = 1, that is, 2x + y - 6 = 0