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Maths-

General

Easy

Question

The line $fraction numerator x over denominator a end fraction$ + $fraction numerator y over denominator b end fraction$ = 1 touches the curve $y space equals space b e to the power of negative x over a end exponent$ at the point :

$(a, \frac{b}{a})$
$(- a, \frac{b}{a})$
$(a, \frac{a}{b})$
None of these

hint Hint:

We have to find the point of intersection of line and curve, the point where they intersect must have common tangent. Find the slope which will also be common for both.

The correct answer is: None of these

$rightwards double arrow x over a space plus space y over b space equals space 1 D i f f e r e n t i a t i n g space t h e space a b o v e space e q u a t i o n space w r t space d x space w e space g e t rightwards double arrow 1 over a space plus space 1 over b fraction numerator d y over denominator d x end fraction space equals space 0 rightwards double arrow fraction numerator d y over denominator d x end fraction equals space minus b over a rightwards double arrow y space equals space b e to the power of negative x over a end exponent D i f f r e n t i a t i n g space t h e space a b o v e space e q u a t i o n space w e space g e t rightwards double arrow fraction numerator d y over denominator d x end fraction space equals space b e to the power of negative bevelled x over a end exponent space cross times negative 1 over a rightwards double arrow fraction numerator d y over denominator d x end fraction equals space minus b over a e to the power of negative x over a end exponent S i n c e space t h e y space b o t h space h a v e space c o m m o n space tan g e n t space e q u a t i n g space t h e i r space s l o p e s rightwards double arrow negative b over a space equals space space minus b over a e to the power of negative x over a end exponent rightwards double arrow e to the power of negative x over a end exponent space equals space 1 space rightwards double arrow space 1 over e to the power of begin display style x over a end style end exponent space equals 1 rightwards double arrow e to the power of x over a end exponent space equals space 1$
$T a k i n g space log space o n space b o t h space s i d e s rightwards double arrow log open parentheses e to the power of x over a end exponent close parentheses space equals space log space 1 space equals space 0 rightwards double arrow x over a space equals space 0 rightwards double arrow x equals space 0 S u b s t i t u t e space t h e space v a l u e space space o f space x space i n space a n y space e q u a t i o n rightwards double arrow 0 space plus space y over b space equals space 1 rightwards double arrow y space equals space b T h u s space t h e space p o i n t s space a r e space left parenthesis 0 comma b right parenthesis$

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