## Calculus 8th Edition

By rotating and translating the parabola , we can assume that it has an equation of the form as $y=cx^{2}$ 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.