Hi guys
First post here so I don't know how I'm supposed to approach this site, but basically I'm having some (possibly stupid) trouble with a bit of basic algebra in a calculus question:
The Question
Consider drawing some straight lines to form a pattern. Join a line from (0.1,0) to the point (0,0.9), then join (0.2,0) to (0,0.8), etc. In general, consider joining the points (a,0) to (0,b), where a+b=1 and 0<a, b<1.
In this question we hope to find that all these lines are tangent to the curve at sqrt(x) +sqrt(y) = 1 (with 0<x, y<1)
From the first and second parts to this question, I learn that bx+ay=ab when joining (a,0) and (b,0). I also learn that the derivative to this expression is -sqrt(y/x)
Part c, however, asks me to "Let (p,q) be a point on the curve sqrt(x) + sqrt(y) = 1. Show that the tangent line to this curve at (p,q) has the equation
x*sqrt(q)+y*sqrt(p)=sqrt(pq)
Then, explain why this line, as well as the line from part a) (bx+ay=ab) are the same."
Now I didn't have any trouble proving part a or b, but when I got to c, I found that
x*sqrt(q) + y*sqrt(p) = p*sqrt(q) + q*sqrt(p)
Instead of sqrt(pq)
This answer was obtained by using the linear formula. I knew that the gradient of this line had to be -sqrt(q/p) (carrying on from part b). I also knew that the point (p,q) lay somewhere on this curve.
Using this knowledge, I solved using the formula y-y1 = m(x-x1). giving me the result mentioned above.
I know that this must lie on the curve sqrt(x)+sqrt(y)=1, so therefore we have sqrt(p)+sqrt(q)=1. My problem is in rearranging x*sqrt(q) + y*sqrt(p) = p*sqrt(q) + q*sqrt(p) to give x*sqrt(q)+y*sqrt(p)=sqrt(pq)... and for some reason, my algebra has failed me!!
If anyone knows where I'm going wrong, and how I can approach this probably simple problem, your help will be much appreciated.
First post here so I don't know how I'm supposed to approach this site, but basically I'm having some (possibly stupid) trouble with a bit of basic algebra in a calculus question:
The Question
Consider drawing some straight lines to form a pattern. Join a line from (0.1,0) to the point (0,0.9), then join (0.2,0) to (0,0.8), etc. In general, consider joining the points (a,0) to (0,b), where a+b=1 and 0<a, b<1.
In this question we hope to find that all these lines are tangent to the curve at sqrt(x) +sqrt(y) = 1 (with 0<x, y<1)
From the first and second parts to this question, I learn that bx+ay=ab when joining (a,0) and (b,0). I also learn that the derivative to this expression is -sqrt(y/x)
Part c, however, asks me to "Let (p,q) be a point on the curve sqrt(x) + sqrt(y) = 1. Show that the tangent line to this curve at (p,q) has the equation
x*sqrt(q)+y*sqrt(p)=sqrt(pq)
Then, explain why this line, as well as the line from part a) (bx+ay=ab) are the same."
Now I didn't have any trouble proving part a or b, but when I got to c, I found that
x*sqrt(q) + y*sqrt(p) = p*sqrt(q) + q*sqrt(p)
Instead of sqrt(pq)
This answer was obtained by using the linear formula. I knew that the gradient of this line had to be -sqrt(q/p) (carrying on from part b). I also knew that the point (p,q) lay somewhere on this curve.
Using this knowledge, I solved using the formula y-y1 = m(x-x1). giving me the result mentioned above.
I know that this must lie on the curve sqrt(x)+sqrt(y)=1, so therefore we have sqrt(p)+sqrt(q)=1. My problem is in rearranging x*sqrt(q) + y*sqrt(p) = p*sqrt(q) + q*sqrt(p) to give x*sqrt(q)+y*sqrt(p)=sqrt(pq)... and for some reason, my algebra has failed me!!
If anyone knows where I'm going wrong, and how I can approach this probably simple problem, your help will be much appreciated.
