Question

The tangent to the curve xy=4 at a point p in the first quadrant meets the x-axis at A and the y-axis at B. Prove that the area of triangle AOB, where O is the origin, is independent of the position P. Really need help with this question!!! D:

Answer 1

I think this is how it would look like but im not sure

Answer 2

is the area always 1?

Answer 3

It doesnt say

Answer 4

No, i mean the area is always 2

well, suppose P = (a,1/a), then the tangent line at x = a is
y = (-1/a^2) (x - a) + 1/a

when x = 0, y = 2/a
when y = 0, x = 2a

A = (1/2) (2/a) (2a) = 2

so it doesn't depends on the coordinate of P

Answer 5

hmm...doesnt make sense to me, sorry im lost

Answer 6

I had y = 1/x when I should have had y = 4/x. But the process is still the same

Answer 7

A = (1/2) (base ) (height)= (1/2) (2a)(8/a) = 8

Answer 8

you know how to find the tangent line right?

Answer 9

oh okay that makes sense thanks