Answer
TRUE
Work Step by Step
By rotating and translating the parabola, we can assume that it has an equation of the form $y=cx^{2}$ where $c>0$.
The tangent at the point $(a,ca^{2})$ is the line
$y-ca^{2}=2ca(x-a)y$
$=2cax-ca^{2}$
The tangent meets the parabola at the points $(x,cx^{2})$
where $cx^{2}=2cax-ca^{2}$
Thus,
$x^{2}=2ax-a^{2}$
$x^{2}-2ax+a^{2}=0$
$(x-a)^{2}=0$
This implies $x=a$
Therefore, the tangent meets the parabola at the points $(a,ca^{2})$ at exactly one point.
Hence, the given statement is true.
