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mathstina
 2 years ago
Show that the sum of the x, y, z intercepts of any tangent plane to the surface √x+√y+√z= √c is a constant.
mathstina
 2 years ago
Show that the sum of the x, y, z intercepts of any tangent plane to the surface √x+√y+√z= √c is a constant.

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TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1similar to the last one in approach; get the equation of the tangent plane for a point on the surface P=(p,q,r)

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1i ve get the eqn for tangent plane?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1no, you need to find it you need the gradient at that point (normal to the plane) and a point on the plane

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1\[1/2[1/\sqrt{p} +1/\sqrt{q}+1/\sqrt{r}]=0\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1yes, now dot that with a general vector in the plane to get the equation of the tangent plane

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1actually you should have written that as a vector\[\nabla f(p,q,r)=\frac12\langle p^{1/2},q^{1/2},r^{1/2}\rangle\]

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1\[1/2{p/\sqrt{p}(xp) + q/\sqrt{q}(yq) +r/\sqrt{r}(zr)} =0\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1I think you got an extra p,q and r somehow\[\nabla f(p,q,r)=\frac12\langle p^{1/2},q^{1/2},r^{1/2}\rangle\cdot\langle xp,yq,zr\rangle=0\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[\frac12\langle p^{1/2},q^{1/2},r^{1/2}\rangle\cdot\langle xp,yq,zr\rangle=0\]\[p^{1/2}x+q^{1/2}y+r^{1/2}z=p^{1/2}+q^{1/2}+r^{1/2}\]

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1hw to get the second eqn?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you mean the part on the right? I just simplified the dot product and moved all the constants to one side

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[\frac12\langle p^{1/2},q^{1/2},r^{1/2}\rangle\cdot\langle xp,yq,zr\rangle=0\]\[p^{1/2}xp^{1/2}+q^{1/2}yq^{1/2}+r^{1/2}zr^{1/2}=0\]\[p^{1/2}x+q^{1/2}y+r^{1/2}z=p^{1/2}+q^{1/2}+r^{1/2}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1now plug in for each intercept x=0, y=0, z=0 add the three results, what do you get?

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1what happened to the 1/2 for the dot product?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1all that stuff was =0, so multiplying both sides by 2 it is still =0, but simpler

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[\frac12\langle p^{1/2},q^{1/2},r^{1/2}\rangle\cdot\langle xp,yq,zr\rangle=0\]\[\implies\langle p^{1/2},q^{1/2},r^{1/2}\rangle\cdot\langle xp,yq,zr\rangle=0\]

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1k. hw do i sub for each intercept?

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1is it (x, y,z) =1 respectively

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1x=0 will give you one equation, y=0 another, z=0 another add them all up

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1sorry, I guess I mean xy=0, yz=0, xz=0

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1to find the xintercept, plug in y=0 and z=0 what do you get?

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1r −1/2 z=p 1/2 +q 1/2 +r 1/2

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you plugged in x=0 and y=0, so this is going to be the zintercept solve for z

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1how do you get that?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1where did p and q go?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1besides, what you wrote would imply that z=r

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1but x and y being zero does not change the value of p and q

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1p, q, and r are the coordinates of the point *on the shape* from which we got our tangent plane x, y, and z can be anywhere in the plane, but the equation of the plane still depends on p, q, and r

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1the xintercept *of the plane* will be found by plugging in y=z=0 to our formula *for the plane*

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1xintercept:\[p^{1/2}x=p^{1/2}+q^{1/2}+r^{1/2}\implies x=p+p^{1/2}q^{1/2}+q^{1/2}r^{1/2}\]now find the y and zintercepts

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1sorry, typo\[p^{1/2}x=p^{1/2}+q^{1/2}+r^{1/2}\implies x=p+p^{1/2}q^{1/2}+p^{1/2}r^{1/2}\]

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1q −1/2 y=p 1/2 +q 1/2 +r 1/2 ⟹y=p1/2q1q+p 1/2 r 1/2

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1how to solve for the unknowns?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1you don't, we are trying to show that this is true for *all* points that are on the surface so we have to leave it general. same reason we went ahead with the last one in terms of a,b, and c

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1so get the zintercept, then add the x, y, and zintercept together

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1just add them; it will look ugly at first, but a miracle will happen...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1erm, just add the intercepts...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[x_0=p+p^{1/2}q+p^{1/2}r^{1/2}\]\[y_0=p^{1/2}q^{1/2}+q+q^{1/2}r^{1/2}\]\[z_0=p^{1/2}r^{1/2}+q^{1/2}r^{1/2}+r\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1the sum of the intercepts is then\[x_0+y_0+z_0=p+q+r+2p^{1/2}q^{1/2}+2p^{1/2}r^{1/2}+2q^{1/2}r^{1/2}\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1this looks ugly, but it can be factored look at the original function... what is the relationship?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[p+q+r+2p^{1/2}q^{1/2}+2p^{1/2}r^{1/2}+2q^{1/2}r^{1/2}=(\sqrt p+\sqrt q+\sqrt r)^2\]now do you see how this is connected to the original function?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1how did I know the above you mean? practice and recognizing forms

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1do you see what to do from here at all?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1remember that p, q, and r satisfy our original equation, so\[\sqrt p+\sqrt q+\sqrt r=\sqrt c\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1combine that with the fact that the sum of the intercepts we have shown to be\[(\sqrt p+\sqrt q+\sqrt r)^2=(\sqrt c)^2=c\]and you see that all intercepts for any tangent plane sum to exactly c QED

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1how to get simplified to (p √ +q √ +r √ ) 2 ?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1I did it by observation and recognizing the form, but... look at the intercepts\[x_0=p+p^{1/2}q+p^{1/2}r^{1/2}=p^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})\]\[y_0=p^{1/2}q^{1/2}+q+q^{1/2}r^{1/2}=q^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})\]\[z_0=p^{1/2}r^{1/2}+q^{1/2}r^{1/2}+r=r^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})\]

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1\[x_0+y_0+z_0\]\[=p^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})+q^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})+r^{1/2}(p^{1/2}+q^{1/2}+r^{1/2})\]\[=(p^{1/2}+q^{1/2}+r^{1/2})(p^{1/2}+q^{1/2}+r^{1/2})\]\[=(p^{1/2}+q^{1/2}+r^{1/2})^2\]

mathstina
 2 years ago
Best ResponseYou've already chosen the best response.1ok .thanks a lot!!!!!

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.1welcome, this was a good mental exercise to start my day :)
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