Student Question

`x=t+1 , y=t^2+3t` Find all points (if any) of horizontal and vertical tangency to the curve.

Expert Answers

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Parametric curve (x(t),y(t)) has a horizontal tangent if its slope `dy/dx` is zero i.e. when `dy/dt=0` and `dx/dt!=0`

It has a vertical tangent, if its slope approaches infinity i.e. `dx/dt=0` and `dy/dt!=0`

Given parametric equations are:





For Horizontal tangents,





Corresponding point on the curve can be found by plugging in the value of t in the equations,





Horizontal tangent is at the point `(-1/2,-9/4)`

For vertical tangents,


However `dx/dt=1!=0`

So there are no vertical tangents.

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