Question

Consider the parabola r​(t)equalsleft angle at squared plus 1 comma t right angle​, for minusinfinityless thantless thaninfinity​, where a is a positive real number. Find all points on the parabola at which r and Bold r prime are orthogonal?

Answer 1

Answer:

t=0

Step-by-step explanation:

r=<$at^2$+1,$t$>

by differentiating the r vector component by component

r' = <2at, 1>

Two vector are orthogonal when the dot product between them is zero, so:

r'·r=0

<$at^2$+1,$t$>·<2at, 1>=2$a^2t^3$+2at+t=0

common factor

t($2a^2t^2$+2a+1)=0

Then, the only real value for t is zero.

-> t=0