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In this problem we consider drawing some straight lines which form a nice pattern. Consider joining the point (0.1,0) to the point (0,0.9) by a line segment; then
joining (0.2,0) to (0,0.8) by a line segment; and so on. In general, consider joining the points (a,0) and (0,b) where a + b = 1 and 0 < a, b < 1.

We will show in this problem that all these straight lines are tangent to the curve with equation √x + √y = 1
(with 0 < x, y < 1).

Now i've done A and B but they may be useful as part of the others. I want to find out C and D but am having trouble can someone please help.

(a) Show that the line joining (a, 0) and (0, b), where a + b = 1 and 0 < a, b < 1, has equation
bx + ay = ab
(b) Show that if x and y are related by the equation
√x + √y = 1 then
dy
dx= 􀀀-√y/x
(c) Let (p,q) be a point on the curve √x + √y = 1. Show that the tangent line to this curve at (p,q) has equation x*√q + y√p = √p*q
(d) Explain why the lines considered in parts (a) and (c) are the same, i.e. the lines we originally drew in part (a) are precisely the tangent lines to the curve
√x + √y=1

Thanks.

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Alex

  1. Hi again. Just realised that i got B wrong. So it means i need that too. Just to clarify (b) Show that if x and y are related by the equation
    √x + √y = 1 then
    dy _____
    dx= 􀀀-√(y/x)
    _ _
    I got -√y/-√x.

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    Alex

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