Application of Derivatives

Find the equation of tangents to the curve y = cos (x + y), – 2 ≤ x ≤ 2 that are parallel to the line x + 2y = 0. - Wired Faculty

Question

Wired Faculty App

Find the equation of tangents to the curve
y = cos (x + y), – 2 $straight pi$ ≤ x ≤ 2 $straight pi$
that are parallel to the line x + 2y = 0.

Solution

The equation of curve is
y = cos (x + y) ...(1)
Differentiating both sides w.r.t.x, we get,
   $dy over dx space equals space minus sin space left parenthesis straight x plus straight y right parenthesis. space space space open parentheses 1 plus dy over dx close parentheses$
or $dy over dx space equals space minus sin space left parenthesis straight x plus straight y right parenthesis space minus space sin space left parenthesis straight x plus straight y right parenthesis. space dy over dx$
$therefore space space space left square bracket 1 plus sin space left parenthesis straight x plus straight y right parenthesis space dy over dx space equals space minus sin space left parenthesis straight x plus straight y right parenthesis therefore space space space space space space space space dy over dx space equals space minus fraction numerator sin space left parenthesis straight x plus straight y right parenthesis over denominator 1 plus sin space left parenthesis straight x plus straight y right parenthesis end fraction$
$therefore space space space slope space of space tangent space at space left parenthesis straight x comma space straight y right parenthesis space space equals space minus fraction numerator sin space left parenthesis straight x plus straight y right parenthesis over denominator 1 plus sin space left parenthesis straight x plus straight y right parenthesis end fraction Consider space the space line space space space straight x plus 2 straight y space equals space 0 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... left parenthesis 2 right parenthesis Its space slope space space equals space minus 1 half Since space lines space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space are space parallel. therefore space space space space their space slopes space are space equal. space space space space therefore space space space space space minus fraction numerator sin space left parenthesis straight x plus straight y right parenthesis over denominator 1 plus sin space left parenthesis straight x plus straight y right parenthesis end fraction space equals space minus 1 half space space space space space space therefore space space space space space space space 2 space sin space left parenthesis straight x plus straight y right parenthesis space equals space 1 plus sin left parenthesis straight x plus straight y right parenthesis space space space space space space space therefore space space space space space sin space left parenthesis straight x plus straight y right parenthesis space equals space 1 space space space space space space space or space space space space sin space left parenthesis straight x plus straight y right parenthesis space equals space sin space space straight pi over 2 space space space space space space therefore space space space space space space space space straight x plus straight y space equals space nπ plus left parenthesis negative 1 right parenthesis to the power of straight n straight pi over 2 comma space space space space straight n element of straight Z space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma space space space space straight y space equals space cos space open square brackets straight n space straight pi space plus space left parenthesis negative 1 right parenthesis to the power of straight n straight pi over 2 close square brackets comma space space space straight n element of straight Z Now comma space space space space space space space minus 2 straight pi space less or equal than straight x less or equal than 2 straight pi therefore 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 straight x equals negative fraction numerator 3 straight pi over denominator 2 end fraction comma space space straight pi over 2$ $therefore space space space space space space tangents space to space the space given space curve space left parenthesis 1 right parenthesis space are space parallel space to space the space line space straight x plus 2 space straight y space equals space 0 space at space points space open parentheses negative fraction numerator 3 straight pi over denominator 2 end fraction comma space 0 close parentheses comma space open parentheses straight pi over 2 comma space 0 close parentheses$
The equation of tangent at $open parentheses negative fraction numerator 3 straight pi over denominator 2 end fraction comma space space 0 close parentheses$ is
$straight y minus 0 space equals space minus 1 half open parentheses straight x plus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses space space space or space space space space space 2 straight y space equals negative straight x minus fraction numerator 3 straight pi over denominator 2 end fraction$
or    $4 straight y space equals space minus 2 straight x plus straight pi space space space space or space space space 2 straight x plus 4 straight y minus straight pi space equals space 0$
The equation of tangent at $open parentheses straight pi over 2 comma space 0 close parentheses$ is
   $straight y minus 0 space equals space minus 1 half open parentheses straight x minus straight pi over 2 close parentheses space space space space or space space space 2 straight y space equals negative straight x plus straight pi over 2$
or $4 straight y space equals space minus 2 straight x plus straight pi space space space space space space space or space space space space space 2 straight x plus 4 straight y minus straight pi space equals space 0$

For Daily Free Study Material Join wiredfaculty