Consider the curve $y=x^4$.
$(A)$ – The item $A$ was asked yesterday, I put it here in case it is useful.
$(B)$ – Determine the equations of the tangent lines to the curve that pass through the point $(2,0)$.
I don't understand what I should do in this question. There is actually no tangent line to the curve that passes through $(2,0)$. I thought that it could be about translating the line tangent to the curve, but it also doesn't seems to make sense.
Best Answer
There is no tangent line to that curve at $(2,0)$, but there are two tangent lines to that curve that pass through $(2,0)$. It is tangent to the curve at some point $(a,a^4)$, were the slope is $4a^3$. So $$ 4a^3 = \text{slope} = \frac{a^4-0}{a-2}, $$ whence we get $$ 4a^3(a-2) = a^4. $$ If $a\ne0$ we can divide both sides of this by $a^3$, getting $4(a-2)=a$. It follows that $a=8/3$. On the other hand, if $a=0$, we get another such tangent line.