If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joinin…

Question

If the tangent to the curve y = x + sin y at a point

(a, b) is parallel to the line joining $$\left( {0,{3 \over 2}} \right)$$ and $$\left( {{1 \over 2},2} \right)$$, then :

Accepted Answer

Slope of tangent to the curve y = x + siny

at (a, b) is = $${{2 - {3 \over 2}} \over {{1 \over 2} - 0}}$$ = 1

Given curve y = x + sin y

$$ \Rightarrow $$ $${{dy} \over {dx}} = 1 + \cos y.{{dy} \over {dx}}$$

$$ \Rightarrow $$ (1 - cos y) ({{dy} \over {dx}}\] = 1

$$ \Rightarrow $$ $${\left. {{{dy} \over {dx}}} \right|_{\left( {a,b} \right)}}$$ = $${1 \over {1 + \cos b}}$$

Now according to question, $${1 \over {1 + \cos b}} = 1$$

$$ \Rightarrow $$ cos b = 0

$$ \Rightarrow $$ sin b = $$ \pm $$ 1

Point (a, b) lies on curve y = x + sin y

$$ \therefore $$ b = a + sin b

$$ \Rightarrow $$ |b - a| = |sin b| = 1

Solutions · 1