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Maths-

General

Easy

Question

If the curves $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator 12 end fraction equals 1 blank$ and $y to the power of 3 end exponent equals 8 x blank$ intersect at right angle then the value of $a to the power of 2 end exponent$ is equal to

16
12
8
4

The correct answer is: 4

Given curves are $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator 12 end fraction equals 1$
$rightwards double arrow blank fraction numerator 2 x over denominator a to the power of 2 end exponent end fraction plus fraction numerator 2 y over denominator 12 end fraction bullet fraction numerator d y over denominator d x end fraction equals 0$

$rightwards double arrow blank fraction numerator d y over denominator d x end fraction equals negative fraction numerator 12 x over denominator a to the power of 2 end exponent y end fraction equals m subscript 1 end subscript$ (say)

And $y to the power of 3 end exponent equals 8 x rightwards double arrow 3 y to the power of 2 end exponent fraction numerator d y over denominator d x end fraction equals 8$

$rightwards double arrow blank fraction numerator d y over denominator d x end fraction equals fraction numerator 8 over denominator 3 y to the power of 2 end exponent end fraction equals m subscript 2 end subscript$ (say)

For $theta equals fraction numerator pi over denominator 2 end fraction comma 1 plus m subscript 1 end subscript m subscript 2 end subscript equals 0$

$rightwards double arrow blank 1 plus open parentheses fraction numerator negative 12 x over denominator a to the power of 2 end exponent y end fraction close parentheses open parentheses fraction numerator 8 over denominator 3 y to the power of 2 end exponent end fraction close parentheses equals 0$

$rightwards double arrow blank 3 a to the power of 2 end exponent open parentheses 8 x close parentheses minus 96 x equals 0$

$rightwards double arrow blank a to the power of 2 end exponent equals 4$

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