Maths-
General
Easy

Question

The sum of the subtangent and subnormal to the curve  at x equals c left square bracket 2 cos invisible function application theta minus log invisible function application left parenthesis cos invisible function application e c theta plus cot invisible function application theta right parenthesis right square bracket comma y equals c sin invisible function application 2 theta text  at  end text theta equals pi over 3 is

  1. c
  2. 2 c
  3. 3 c
  4. 5 c

Hint:

We are given the equations of x and y. We are also given the value of θ. We have to find the sum of length of subnormal and length of subtangent at given value of θ.

The correct answer is: 2 c


    x = c[2cosθ - log(cosecθ + cotθ)
    y = csin2θ
    A tangent to the curve is given by the slope at the point. The slope is given by fraction numerator d y over denominator d x end fraction
    let's differentiate x w.r.t θ.
    x equals c left square bracket 2 cos theta minus log left parenthesis cos e c theta plus c o t theta right parenthesis
fraction numerator d x over denominator d theta end fraction space equals space c left square bracket 2 left parenthesis negative sin theta right parenthesis space minus space fraction numerator 1 over denominator cos e c theta space plus space c o t theta end fraction fraction numerator d over denominator d theta end fraction cos e c theta space plus c o t theta
space space space space space space space space equals c left square bracket negative 2 sin theta space minus space fraction numerator left parenthesis negative cos e c theta c o t theta space minus cos e c squared theta right parenthesis over denominator cos e c theta space plus space c o t theta end fraction
space space space space space space space equals c left square bracket negative 2 sin theta space minus left parenthesis negative cos e c theta right parenthesis fraction numerator c o t theta space plus space cos e c theta over denominator cos e c theta space plus space c o t theta end fraction right square bracket
space space space space space space equals c left square bracket negative 2 sin theta space plus space cos e c theta right square bracket

space space spaceWe will differentiate y w.r.t θ
    y equals c sin 2 theta
fraction numerator d y over denominator d theta end fraction space equals space c cos 2 theta space fraction numerator d over denominator d theta end fraction 2 theta
space space space space space space space space equals 2 c cos 2 theta
    Now, let us denote slope by mm space equals fraction numerator d y over denominator d x end fraction
space space space space space equals space fraction numerator d y over denominator d theta end fraction cross times fraction numerator d theta over denominator d x end fraction
space space space space space equals 2 c cos 2 theta space cross times space fraction numerator 1 over denominator c left square bracket negative 2 sin theta space plus space cos e c theta right square bracket end fraction
left parenthesis m right parenthesis subscript theta equals pi over 3 space end subscript equals 2 c cos left parenthesis fraction numerator 2 pi over denominator 3 end fraction right parenthesis space cross times space fraction numerator 1 over denominator c left square bracket negative 2 sin begin display style pi over 3 end style plus cos e c begin display style straight pi over 3 end style right square bracket end fraction
space space space space space space space space space space space space space space space equals fraction numerator 2 c left parenthesis negative begin display style 1 half end style right parenthesis over denominator c left parenthesis negative 2 cross times begin display style fraction numerator square root of 3 over denominator 2 end fraction end style plus begin display style fraction numerator 2 over denominator square root of 3 end fraction end style right parenthesis space space space space space space space space space end fraction space space space... left curly bracket cos fraction numerator 2 pi over denominator 3 end fraction equals cos left parenthesis pi minus straight pi over 3 right parenthesis equals negative 1 half right curly bracket
space space space space space space space space space space space space space space equals fraction numerator negative 1 over denominator negative square root of 3 plus begin display style fraction numerator 2 over denominator square root of 3 end fraction end style end fraction
space space space space space space space space space space equals space space minus fraction numerator square root of 3 over denominator negative 3 space plus space 2 end fraction
space space space space space space
space space space

space space space space space space
    m = √3
    y=(y)θ=π/3c sin fraction numerator 2 pi over denominator 3 end fraction
    y subscript 1 equals c sin pi over 3 space space space... left curly bracket sin fraction numerator 2 pi over denominator 3 end fraction space equals sin left parenthesis pi minus pi over 3 right parenthesis space equals sin pi over 3 right curly bracket
space space space space equals c space cross times space fraction numerator square root of 3 over denominator 2 end fraction
space space space space equals fraction numerator square root of 3 over denominator 2 end fraction c
    L e n g t h space o f space s u b n o r m a l equals open vertical bar y subscript 1 m close vertical bar
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar fraction numerator square root of 3 over denominator 2 end fraction c space cross times space square root of 3 close vertical bar

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar 3 over 2 c close vertical bar
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 3 over 2 c

space
    L e n g t h space o f space s u b tan g e n t space equals open vertical bar y subscript 1 over m close vertical bar
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open vertical bar fraction numerator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style c over denominator square root of 3 end fraction close vertical bar
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half c
    L e n g t h space o f space s u b tan g e n t space plus space l e n g t h space o f space s u b n o r m a l space equals fraction numerator 3 c over denominator 2 end fraction plus fraction numerator 1 c over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 4 c over denominator 2 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 cSo, the answer is 2c

    For such questions, we should know the formula of subtangent and subnormal

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