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Maths-

General

Easy

Question

The line $left parenthesis x divided by a right parenthesis plus left parenthesis y divided by b right parenthesis equals 2$ , touches the curve $fraction numerator x to the power of n end exponent over denominator a to the power of n end exponent end fraction plus fraction numerator y to the power of n end exponent over denominator b to the power of n end exponent end fraction equals 2$ , at

$(b, a)$
$(- b, - a)$
$(a, b)$
None of these

The correct answer is: $left parenthesis a comma blank b right parenthesis$

Let the point of contact be $left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$
The equation of curve is

$fraction numerator x to the power of n end exponent over denominator a to the power of n end exponent end fraction plus fraction numerator y to the power of n end exponent over denominator b to the power of n end exponent end fraction equals 2$

$rightwards double arrow blank open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis end subscript equals negative fraction numerator b to the power of n end exponent over denominator a to the power of n end exponent end fraction blank. fraction numerator x subscript 1 end subscript superscript n minus 1 end superscript over denominator y subscript 1 end subscript superscript n minus 1 end superscript end fraction$ ...(i)

The equation of line is $fraction numerator x over denominator a end fraction plus fraction numerator y over denominator b end fraction equals 2$

$rightwards double arrow blank open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis end subscript equals negative fraction numerator b over denominator a end fraction$ ...(ii)

From Eqs. (i) and (ii), we get

$negative fraction numerator b over denominator a end fraction equals negative fraction numerator b to the power of n end exponent over denominator a to the power of n end exponent end fraction blank. fraction numerator x subscript 1 end subscript superscript n minus 1 end superscript over denominator y subscript 1 end subscript superscript n minus 1 end superscript end fraction blank rightwards double arrow blank x subscript 1 end subscript equals fraction numerator a y subscript 1 end subscript over denominator b end fraction$ ...(iii)

Also, $left parenthesis x subscript 1 end subscript comma blank y subscript 1 end subscript right parenthesis$ lies on the given line

$therefore blank fraction numerator a y subscript 1 end subscript over denominator a b end fraction plus fraction numerator y subscript 1 end subscript over denominator b end fraction equals 2$

$rightwards double arrow blank y subscript 1 end subscript equals b$ and $x subscript 1 end subscript equals a$

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