Tangents lines passing through $(0,0)$ of $x=y^2-y+1$

Question

What are the equations of the tangents lines passing through $(0,0)$ of $x=y^2-y+1$? How do you get them?

Answer 1

Hint:

Consider a variable straight line through the origin: $y=tx$, and write the condition which ensures the quadratic equation in $x$ for the intersection points with the parabola has a double root.

Answer 2

Hint:

Let $(a,b)$ be the point on the parabola.

Slope of line through $(a,b)$ and $(0,0):$

$m= \dfrac {b}{a} .$

Differentiate the equation:

$1= 2y\dfrac{dy}{dx} -\dfrac{dy}{dx}.$

$\dfrac{dy}{dx} = \dfrac{1}{2y-1}.$

Equate: $m= \dfrac{dy}{dx}$ at $(a,b).$

$\dfrac{1}{2b-1}= \dfrac{b}{a}.$

Use $a = b^2 -b +1$ to solve for $b$ in the above equation.

Can you take it from here?