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A line through the point P(2,2) cuts the x and y axes at A and B, respectively. Find the minimum length of line segment AB.

Sorry about the fact that this a few days to answer,

but other matters gave me other stuff to do.

The equation of such a line would be y-2 = m(x-2).

As can be seen, (2,2) would be a point on that line.

Rewritten, this is y = mx - 2m + 2.

If it crosses the x-axis at x = A, the point is (A,0).

This makes the line pass through (2,2) and (A,0).

This makes the slope m = (2-0)/(2-A) = 2/(2-A).

Putting that back in the equation gives y = 2x/(2-A) - 2(2/(2-A)) + 2.

The last two terms combine to -4/(2-A) + (4-2A)/(2-A) = -2A/(2-A) = 2A/(A-2).

That makes the line be y = 2x/(2-A) + 2A/(A-2).

As seen from this, when x-0, y = 2A/(A-2).

It is known the the distance squared is D²(A) = A² +[2A/(A-2)]².

That is, D²(A) = A² + 4A²/(A-2)².

Taking the derivative gives d(D²)/dA = 2A - [8A(A-2)² - 8A²(A-2)]/(A-2)^4.

Cancelling an A-2 on the top and bottom of the 2nd term gives

d(D²)/dA = 2A - [8A(A-2) - 8A²]/(A-2)³.

Noting that 8A(A-2) = 8A²-16A, we can subtract off 8A² and get -16A.

This makes d(D²)/dA = 2A - 16A/(A-2)².

Solving 0 = 2A - 16A/(A-2)² leads to adding 16A/(A-2)² to both sides.

This gives 16A/(A-2)² = 2A.

Multiplying both sides by (A-2)³/(2A) gives 8 = (A-2)³.

Since 8 = 2³, that means that 2 = A - 2, or A = 4.

As can be seen, if the line crosses the x-axis at A = 4, the slope is -1.

This means it crosses the y-axis at 4 as well.

The smallest area, then, is right triangle with both angles (besides the right angle) 45°.

Calculus

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