Answer to Question #268954 in Calculus for Tithi

Question #268954

If l,m, be the portions of the axes of x and y intercepted by the tangent at any point x, y on the curve (x/a)^2/3+


(y/b)^2/3=1 , show that (l)^2/(a)^2 + (m)^2/(b)^2 =1

1
Expert's answer
2021-11-22T18:47:33-0500


xa23+yb23=1\frac{x}{a}^{\frac{2}{3}}+\frac{y}{b}^{\frac{2}{3}}=1


(23(xa13)×1a+23(yb13)×1b)dydx=0(\frac{2}{3}(\frac{x}{a}^{\frac{-1}{3}})\times\frac{1}{a}+\frac{2}{3}(\frac{y}{b}^{\frac{-1}{3}})\times\frac{1}{b})\frac{dy}{dx}=0


(yb13)×1b)dydx=(xa13)×1a(\frac{y}{b}^{\frac{-1}{3}})\times\frac{1}{b})\frac{dy}{dx}=-(\frac{x}{a}^{\frac{-1}{3}})\times\frac{1}{a}


dydx=ba(xayb)13\frac{dy}{dx}=-\frac{b}{a}(\frac{\frac{x}{a}}{\frac{y}{b}})^-\frac{1}{3}


== (ba)23(xy)13(-\frac{b}{a})^{\frac{2}{3}}(\frac{x}{y})^{\frac{-1}{3}}


let A(x1,y1)    m=(ba)23(xy)13let\ A(x_1,y_1)\implies m=(-\frac{b}{a})^{\frac{2}{3}}(\frac{x}{y})^{\frac{-1}{3}}


Equation of the tangent is

yy1=m(xx1)y-y_1=m(x-x_1)


yy1=(ba)23(xy)13(xx1)y-y_1=(-\frac{b}{a})^{\frac{2}{3}}(\frac{x}{y})^{\frac{-1}{3}}(x-x_1)


open by cross multiplying


a23x113ya23x13y1=b23y113x1+b23y113x1a^{\frac{2}{3}}x_1^{\frac{1}{3}}y-a^{\frac{2}{3}}x^{\frac{1}{3}}y_1=-b^{\frac{2}{3}}y_1^{\frac{1}{3}}x_1+b^{\frac{2}{3}}y_1^{\frac{1}{3}}x_1


(b23y113)x+(a23x13)y=x113y113[a23y123+b23x23](b^{\frac{2}{3}}y_1^{\frac{1}{3}})x+(a^{\frac{2}{3}}x^{\frac{1}{3}})y=x_1^{\frac{1}{3}}y_1^{\frac{1}{3}}[a^{\frac{2}{3}}y_1^{\frac{2}{3}}+b^{\frac{2}{3}}x^{\frac{2}{3}}]


x113y123a23b23[x1a23+y1b23]..................equqation 1x_1^{\frac{1}{3}}y_1^{\frac{2}{3}}a^{\frac{2}{3}}b^{\frac{2}{3}}[\frac{x_1}{a}^{\frac{2}{3}}+\frac{y_1}{b}^{\frac{2}{3}}]..................equqation \ 1


xa23+yb23=1\frac{x}{a}^{\frac{2}{3}}+\frac{y}{b}^{\frac{2}{3}}=1


equation 1    (b23y113)x+(a23x13)y=x113y123a23b23\implies(b^{\frac{2}{3}}y_1^{\frac{1}{3}})x+(a^{\frac{2}{3}}x^{\frac{1}{3}})y=x_1^{\frac{1}{3}}y_1^{\frac{2}{3}}a^{\frac{2}{3}}b^{\frac{2}{3}}


xx113a23+yy113b23=1\frac{x}{x_1^{\frac{1}{3}}a^{\frac{2}{3}}}+\frac{y}{y_1^{\frac{1}{3}}b^{\frac{2}{3}}}=1




as per given a=I b=M


I=x113a23I=x_1^{\frac{1}{3}}a^{\frac{2}{3}}


M=y113b23y_1^{\frac{1}{3}}b^{\frac{2}{3}}


hence

x1a23+y1b23=1\frac{x_1}{a}^{\frac{2}{3}}+\frac{y_1}{b}^{\frac{2}{3}}=1










Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment