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Question

The equation of the one of the tangents to the curve $y equals cos invisible function application left parenthesis x plus y right parenthesis comma blank minus 2 pi less or equal than x less or equal than 2 pi$ that is parallel to the line $x plus 2 y equals 0$ , is

$x + 2 y = 1$
$x + 2 y = \frac{π}{2}$
$x + 2 y = \frac{π}{4}$
None of these

The correct answer is: $x plus 2 blank y equals fraction numerator pi over denominator 2 end fraction$

We have,

$y equals cos invisible function application left parenthesis x plus y right parenthesis$

$rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative sin invisible function application open parentheses x plus y close parentheses open parentheses 1 plus fraction numerator d y over denominator d x end fraction close parentheses$ …(i)

Since the tangent is parallel to $x plus 2 blank y equals 0$

$therefore$ Slope of the tangent $equals negative fraction numerator 1 over denominator 2 end fraction$

$rightwards double arrow fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator 2 end fraction$

Putting $fraction numerator d y over denominator d x end fraction equals negative fraction numerator 1 over denominator 2 end fraction$ in (i), we get

$negative fraction numerator 1 over denominator 2 end fraction equals negative sin invisible function application open parentheses x plus y close parentheses open parentheses 1 minus fraction numerator 1 over denominator 2 end fraction close parentheses$

$rightwards double arrow sin invisible function application left parenthesis x plus y right parenthesis equals 1$ ….(ii)

Now,

$y equals cos invisible function application open parentheses x plus y close parentheses$ and $1 equals sin invisible function application left parenthesis x plus y right parenthesis$

$rightwards double arrow y to the power of 2 end exponent plus 1 equals 1 rightwards double arrow y to the power of 2 end exponent equals 0 rightwards double arrow y equals 0$

Putting $y equals 0$ in $y equals cos invisible function application left parenthesis x plus y right parenthesis$ and $sin invisible function application left parenthesis x plus y right parenthesis equals 1$ , we get

$sin invisible function application x equals 1$ and $cos invisible function application x equals 0 rightwards double arrow x equals negative fraction numerator pi over denominator 2 end fraction comma blank minus fraction numerator 3 pi over denominator 2 end fraction$

Thus, the points on the curve $y equals cos invisible function application left parenthesis x plus y right parenthesis$ where tangents are parallel to $x plus 2 y equals 0$ are $open parentheses pi divided by 2 comma blank 0 close parentheses comma blank open parentheses negative 3 pi divided by 2 comma blank 0 close parentheses$

The equation of the tangent at $open parentheses pi divided by 2 comma blank 0 close parentheses$ is

$y minus 0 equals negative fraction numerator 1 over denominator 2 end fraction open parentheses x minus fraction numerator pi over denominator 2 end fraction close parentheses rightwards double arrow x plus 2 y equals fraction numerator pi over denominator 2 end fraction$

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