Application of Derivatives

Question

At what points of the ellipse 16x² + 9y² = 400, does the ordinate decrease at the same rate at which the abscissa increases?

Solution

Here,   16x² + 9y² = 400 ...(1)
Differentiating both sides w.r.t. 't', we get,
   $32 straight x dx over dt plus 18 straight y dy over dt space equals space 0 space space space or space space 9 straight y dy over dt space equals space minus 16 straight x space dx over dt space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$
But $negative dy over dt space equals space dx over dt$
$therefore space space space space from space space left parenthesis 2 right parenthesis comma space space space minus 9 straight y dx over dt space equals space minus 16 straight x dx over dt therefore space space space space space space 9 straight y space equals space 16 straight x space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight y space equals space 16 over 9 straight x space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis$
Putting $straight y space equals space 16 over 9 straight x space in space left parenthesis 1 right parenthesis comma space we space get comma$
   $16 straight x squared plus 9.256 over 81 straight x squared space equals space space 400$
$therefore space space space space space straight x squared plus 16 over 9 straight x squared space equals space 25 space space space space space space space space space space space space space space space rightwards double arrow space space space space space fraction numerator 25 straight x squared over denominator 9 end fraction space equals space 25 space space space space space space rightwards double arrow space space space straight x squared space equals space 9 space space space space rightwards double arrow space space space straight x space equals negative 3 comma space space 3 therefore space space space space space space from space left parenthesis 3 right parenthesis comma space space space straight y space equals space 16 over 9 left parenthesis negative 3 right parenthesis comma space space 16 over 9 left parenthesis 3 right parenthesis space equals space fraction numerator negative 16 over denominator 3 end fraction comma space space space space 16 over 3 therefore space space space space space space required space points space are space space open parentheses negative 3 comma space space fraction numerator negative 16 over denominator 3 end fraction close parentheses comma space space space space open parentheses 3 comma space space 16 over 3 close parentheses$

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