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Maths-

General

Easy

Question

The equation of the tangent to the curve $x equals 2 cos to the power of 3 end exponent invisible function application theta$ and $y equals 3 sin to the power of 3 end exponent invisible function application theta$ at the point, $theta equals pi divided by 4$ is

$2 x + 3 y = 3 \sqrt{2}$
$2 x - 3 y = 3 \sqrt{2}$
$3 x + 2 y = 3 \sqrt{2}$
$3 x - 2 y = 3 \sqrt{2}$

The correct answer is: $3 x plus 2 y equals 3 square root of 2$

Given curves are $x equals 2 cos to the power of 3 end exponent invisible function application theta$ and $y equals 3 sin to the power of 3 end exponent invisible function application theta$
$therefore blank fraction numerator d x over denominator d theta end fraction equals negative 6 cos to the power of 2 end exponent invisible function application theta sin invisible function application theta$ ,

$fraction numerator d y over denominator d theta end fraction equals 9 sin to the power of 2 end exponent invisible function application theta cos invisible function application theta$

$therefore blank fraction numerator d y over denominator d x end fraction equals negative fraction numerator 9 sin to the power of 2 end exponent invisible function application theta cos invisible function application theta over denominator 6 cos to the power of 2 end exponent invisible function application theta sin invisible function application theta end fraction equals negative fraction numerator 3 over denominator 2 end fraction tan invisible function application theta$

At $theta equals fraction numerator pi over denominator 4 end fraction comma blank fraction numerator d y over denominator d x end fraction equals negative fraction numerator 3 over denominator 2 end fraction tan invisible function application fraction numerator pi over denominator 4 end fraction equals negative fraction numerator 3 over denominator 2 end fraction$

Also, at $theta equals fraction numerator pi over denominator 4 end fraction comma blank x equals 2 cos to the power of 3 end exponent invisible function application fraction numerator pi over denominator 4 end fraction equals negative fraction numerator 1 over denominator square root of 2 end fraction$

And $y equals 3 sin to the power of 2 end exponent invisible function application theta equals 3 open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses to the power of 3 end exponent$

$equals fraction numerator 3 square root of 2 over denominator 4 end fraction$

Equation of tangent at $open parentheses fraction numerator 1 over denominator square root of 2 end fraction comma fraction numerator 3 square root of 2 over denominator 4 end fraction close parentheses$ is

$open parentheses y minus fraction numerator 3 square root of 2 over denominator 4 end fraction close parentheses equals negative fraction numerator 3 over denominator 2 end fraction open parentheses x minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses$

$rightwards double arrow blank 4 y minus 3 square root of 2 equals negative 6 x plus 3 square root of 2$

$rightwards double arrow blank 3 x plus 2 y equals 3 square root of 2$

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