The line $x + ky +k^2 = 0$, where $k$ is a constant, is tangent to curve $y^2 = 4x$ at the point $P$. Find the coordinates of $P$ in terms of $k$.
I solved for $y$ getting $2\sqrt x$ and then substituted this in the equation for tangent to get $x + 2k\sqrt x +k^2 = 0$. Then I set $u = \sqrt x$ to obtain a quadratic equation in terms of $u$ and $k$. Next, I solved for $u$ in terms of $k$ and finally got $x = k^2$ and $y$ as $2k$.
However, the answer in the book says $y$ is $-2k$. How does that make sense?
Best Answer
We are told that $$x=-ky-k^2$$ is tangential to $$x=\frac14y^2$$
We have $$\frac{dx}{dy}=\frac24y$$
$$\frac{2}{4}y=-k$$
Hence $y=-2k$ and $x=\frac14(-2k)^2=k^2$.