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Maths-

General

Easy

Question

If the curves $x to the power of 2 end exponent equals 9 A left parenthesis 9 minus y right parenthesis$ and $x to the power of 2 end exponent equals A left parenthesis y plus 1 right parenthesis$ intersect orthogonally, then the value of $A$ is

3
4
5
7

The correct answer is: 4

On differentiating given curves respectively
$2 x equals negative 9 A fraction numerator d y over denominator d x end fraction blank rightwards double arrow blank open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript c subscript 1 end subscript end subscript equals negative fraction numerator 2 x over denominator 9 A end fraction$

And $2 x equals A blank fraction numerator d y over denominator d x end fraction$

$rightwards double arrow blank open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript c subscript 2 end subscript end subscript equals fraction numerator 2 x over denominator A end fraction$

For orthogonally,

$open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript c subscript 1 end subscript end subscript open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript c subscript 2 end subscript end subscript equals negative 1$

$rightwards double arrow blank open parentheses negative fraction numerator 2 x over denominator 9 A end fraction close parentheses open parentheses fraction numerator 2 x over denominator A end fraction close parentheses equals negative 1$ ...(i)

From given curves,

$9 A open parentheses 9 minus y close parentheses equals A open parentheses y plus 1 close parentheses blank rightwards double arrow blank y equals 8$

Since, $x to the power of 2 end exponent equals 9 A open parentheses 9 minus y close parentheses$

$rightwards double arrow blank x to the power of 2 end exponent equals 9 A open parentheses 9 minus 8 close parentheses equals 9 A$

From Eq. (i), $fraction numerator 4 x to the power of 2 end exponent over denominator 9 A to the power of 2 end exponent end fraction equals 1$

$rightwards double arrow blank fraction numerator 4.9 blank A over denominator 9 A to the power of 2 end exponent end fraction equals 1 blank rightwards double arrow blank A equals 4$

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