# Question #495d2

Sep 30, 2016

Lets say that we can draw a tangent line to parabola from point $P \left(a , b\right)$. Then, there exists point $Q \left(r , s\right)$ in parabola $f \left(x\right) = {x}^{2}$ which our tangent line goes through. Because $Q \in f \left(x\right)$, it follows that:

$f \left(r\right) = {r}^{2} \implies s = {r}^{2} \implies Q \left(r , {r}^{2}\right)$

The equation of the tangent line is: $y = k x + n$, where $k$ is slope and:

$k = f ' \left(x\right) = 2 x$

Because our tangent line goes through $Q$, it follows that: $k = 2 r$ and if we insert coordinates of $Q$ and $k$ into the equation of the tangent line, we get:

$y = k x + n$
${r}^{2} = 2 r \cdot r + n \implies n = - {r}^{2}$

So, our tangent line is: $y = 2 r x - {r}^{2}$.

We started with assumptation that $P \in y$ (tangent line exists and goes through $P$ and $Q$). Then (insert coordinates of $P$ into the equation of tangent line),

$b = 2 r a - {r}^{2} \implies {r}^{2} - 2 r a + b = 0$

The last equation gives us the answer: is it possible to construct the tangent line from arbitrary point $P$ to parabola $f \left(x\right) = {x}^{2}$? If so, there exists the solution of the quadratic equation ${r}^{2} - 2 r a + b = 0$. Because this is a quadratic equation, we may get one solution, two solutions or no solution based on the coefficients of the equation.

So, let solve it:

${r}^{2} - 2 r a + b = 0 \iff$

$\iff {r}^{2} - 2 r a + {a}^{2} - {a}^{2} + b = 0 \iff$

$\iff {\left(r - a\right)}^{2} - {\left(\sqrt{{a}^{2} - b}\right)}^{2} = 0 \iff$

$\iff \left(r - a - \sqrt{{a}^{2} - b}\right) \left(r - a + \sqrt{{a}^{2} - b}\right) = 0 \iff$

$\iff r - a - \sqrt{{a}^{2} - b} = 0 \vee r - a + \sqrt{{a}^{2} - b} = 0 \iff$

$\iff r = a + \sqrt{{a}^{2} - b} \vee r = a - \sqrt{{a}^{2} - b}$

Real solutions exist when ${a}^{2} - b \ge 0$, or $b \le {a}^{2}$.
When ${a}^{2} = b$, we get one solution:

$r = a$.

When ${a}^{2} > b$, we get two solutions:

$r = a + \sqrt{{a}^{2} - b}$
$r = a - \sqrt{{a}^{2} - b}$

Finally, when $b > {a}^{2}$ we don't have solutions.