Implicit domain how to prove #dy/dx# = #-(sqrt(1-y^2))/(sqrt(1-x^2)# ?

If #xsqrt(1-y^2)+ysqrt(1-x^2)=0#

Show that #dy/dx# = #-(sqrt(1-y^2))/(sqrt(1-x^2)#

1 Answer
maganbhai P.
Apr 28, 2018

Please see below.

Explanation:

Here,

#xsqrt(1-y^2)+ysqrt(1-x^2)=0...to(A)#

Let,

#x=sinalpha=>alpha=sin^-1x,where,alphain(-pi/4,pi/4);and#

#y=sinbeta=>beta=sin^-1y,where.betain(-pi/4,pi/4).#

#=>alpha+betain((-pi/4)+(-pi/4),pi/4+pi/4)#

#=>alpha+betain(-pi/2,pi/2)#

So, from #(A)#

#sinalphasqrt(1-sin^2beta)+sinbetasqrt(1-sin^2alpha)=0#

#=>sinalphacosbeta+sinbetacosalpha=0#

#=>sin(alpha+beta)=0#

#=>alpha+beta=sin^-1(0)#

#=>alpha+beta=0#

Subst. back, #alpha and beta#

#=>sin^-1x+sin^-1y=0#

Diff.w.r.t.#x#,

#1/sqrt(1-x^2)+1/sqrt(1-y^2)xx(dy)/(dx)=0#

#1/sqrt(1-y^2)xx(dy)/(dx)=-1/sqrt(1-x^2)#

#=>(dy)/(dx)=(-1/sqrt(1-x^2))/(1/sqrt(1-y^2))#

#=>(dy)/(dx)=-sqrt(1-y^2)/sqrt(1-x^2)#

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