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Question

If $fraction numerator x over denominator a end fraction plus fraction numerator y over denominator b end fraction$ = 1 is a tangent to the curve x = 4t,y = $fraction numerator 4 over denominator t end fraction$ , t $element of$ R then -

a > 0 , b > 0
a > 0, b < 0
a < 0, b > 0
None of these

hint Hint:

Find slope i.e $fraction numerator d y over denominator d x end fraction$

The correct answer is: a > 0 , b > 0

Given : x = 4t, y = $fraction numerator 4 over denominator t end fraction$ , t $element of$ R
$rightwards double arrow x cross times space y space equals space k t space cross times k over t space equals space k squared$ , ( the curve is a rectangular parabola)
$rightwards double arrow fraction numerator x y 1 space plus space y x 1 over denominator 2 end fraction space equals space k squared rightwards double arrow x y 1 space plus y x 1 space equals space 2 k squared$
$rightwards double arrow space L e t space x 1 space equals space k t space a n d space y 1 space equals space k over t rightwards double arrow x cross times k over t space plus space y space cross times k t space equals space 2 k squared rightwards double arrow x over t space plus space t y space space equals space 2 k rightwards double arrow S l o p e space o f space tan g e n t space equals space fraction numerator negative space c o e f f e c i e n t space o f space x over denominator c o e f f e c i e n t space o f space y end fraction rightwards double arrow S l o p e space o f space tan g e n t space equals space fraction numerator begin display style bevelled fraction numerator negative 1 over denominator t end fraction end style over denominator t end fraction space equals fraction numerator negative 1 over denominator t squared end fraction less than space 0$
$rightwards double arrow x over a space plus space y over b space equals space 1 comma space i. e. space tan g e n t space f o r space x y space equals space k squared rightwards double arrow S l o p e space o f space tan g e n t space less than space 0 rightwards double arrow S l o p e space o f space l i n e space equals space fraction numerator begin display style bevelled fraction numerator negative 1 over denominator a end fraction end style over denominator begin display style bevelled 1 over b end style end fraction space less than space 0 rightwards double arrow S l o p e space o f space l i n e space equals space fraction numerator negative b over denominator a end fraction less than space 0 rightwards double arrow S l o p e space o f space l i n e space equals space b over a space greater than space 0 rightwards double arrow T h i s space i m p l i e s space e i t h e r space b space greater than space 0 space comma space a greater than space 0 space o r space b space less than 0 space space a n d space a less than space 0$

$fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction fraction numerator d y over denominator d t end fraction space equals space 4 space a n d space space fraction numerator d x over denominator d t end fraction space equals negative space 4 over t squared S u b s t i t u t i n g space t h e space v a l u e s fraction numerator d y over denominator d x end fraction space equals space fraction numerator begin display style bevelled fraction numerator d y over denominator d t end fraction end style over denominator begin display style bevelled fraction numerator d x over denominator d t end fraction end style end fraction space equals space fraction numerator 4 over denominator begin display style bevelled fraction numerator negative 4 over denominator t squared end fraction end style end fraction space equals space minus 1 over t squared space$

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