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Maths-

General

Easy

Question

At what point on the curve $x to the power of 3 end exponent minus 8 a to the power of 2 end exponent y equals 0$ , the slope of the normal is $negative fraction numerator 2 over denominator 3 end fraction$ ?

$(a, a)$
$(2 a, - a)$
$(2 a, a)$
None of these

The correct answer is: $open parentheses 2 a comma blank a close parentheses$

Given curve is $x to the power of 3 end exponent minus 8 a to the power of 2 end exponent blank y equals 0.$
On differentiating w.r.t. $x$ , we get

$3 x to the power of 2 end exponent minus 8 a to the power of 2 end exponent fraction numerator d y over denominator d x end fraction equals 0$

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

$therefore$ Slope of the normal $equals negative fraction numerator 1 over denominator open parentheses fraction numerator d y over denominator d x end fraction close parentheses end fraction equals negative fraction numerator 1 over denominator fraction numerator 3 x to the power of 2 end exponent over denominator 8 a to the power of 2 end exponent end fraction end fraction equals negative fraction numerator 8 a to the power of 2 end exponent over denominator 3 x to the power of 2 end exponent end fraction$

Given, $fraction numerator negative 8 a to the power of 2 end exponent over denominator 3 x to the power of 2 end exponent end fraction equals fraction numerator negative 2 over denominator 3 end fraction blank rightwards double arrow blank x to the power of 2 end exponent equals 4 a to the power of 2 end exponent blank rightwards double arrow blank x equals plus-or-minus 2 a$

At $x equals plus-or-minus 2 a comma blank y equals plus-or-minus a$

$therefore blank open parentheses x comma blank y close parentheses equals left parenthesis 2 a comma blank a right parenthesis$

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