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Find all the tangents to the curve `y=cos(x+y),-2pilt=xlt=2pi`that are parallel to the line `x+2y=0.` |
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Answer» Correct Answer - `x + 2y = (pi)/(2) and x + 2y = (-3pi)/(2)` Given, `y = cos (x + y)` `rArr ((d y )/(dx)) = - sin (x + y ) * ( 1+ (dy)/(dx))" " ` … (i) Since, tangent is parallel to `x + 2y =0`, then slope `(dy)/(dx) = - (1)/(2)` From Eq. (i), `- (1)/(2) = - sin (x + y ) *( 1- (1)/(2))` `rArr sin (x + y ) = 1, ` which shows `cos ( x+ y ) =0` ` therefore " " y =0` `rArr " " x + y = (pi)/(2) or - ( 3pi)/(2)` `therefore " " x = (pi)/(2) or - ( 3pi)/(2)` Thus, required points are ` ((pi)/(2), 0 ) and ( - ( 3pi) /( 2), 0 )` `therfore ` Equation of tangents are `" " (y -0)/( x - pi//2) = - (1) /(2)` and ` " " ( y - 0 ) /( x + 3pi//2) = -(1) /(2) rArr 2y = - x + (pi) /(2)` and ` " " 2y = - x - ( 3pi) /(2)` `rArr " " x + 2y = (pi)/(2)` and `" "x + 2h = - ( 3pi ) /(2)` are the required equations of tangents. |
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