A curve is defined by the parametric equations x = t^2 and y = sint, for -π<t≤π/2. Find the equation of the normal to the curve that is parallel to the y-axis.Thanks!

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You first need to find the equation of the tangent line to the curve and then you may find the equation of the normal to the curve.

You should remember that the equation of the tangent line at a point `x_1`  is:

`y = f(x_1) + m(x-x_1)`

`m = (dy)/(dx) =gt m = ((dy)/(dt))/((dx)/(dt))`

`m = cos t/(2t)`

For `t = pi/2 =gt m = 0`

For `t = -pi =gt m = -1/(-2pi)`

Hence, evaluating the equation of tangent line at `t=-pi`  yields:

`y = sin (-pi) + m(x + pi)`

`y = 0 + x/(2pi)+ 1/2`

`y =x/(2pi) + 1/2`

You should remember that the normal to the curve is also perpendicular to the tangent line such that:

`m_1*m_2 = -1`

`m_1, m_2`  are the slopes of these lines

`m_2 = -1/(1/(2pi)) = -2pi`

Hence, the equation of normal to the curve at `t=-pi ` is `y=-2pix.`

Sources:

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