algebra precalculus - What equation intersects only once with $f(x)=sqrt{1-(x-2)^2}$

Being $f(x)=\sqrt{1-(x-2)^2}$ I have to know what linear equation only touches the circle once(only one intersection), and passes by $P(0,0)$.

So the linear equation must be $y=mx$ because $n=0$.

I have a system of 2 equations:
$$\begin{align} y&=\sqrt{1-(x-2)^2}\\ y&=mx \end{align}$$

So I equal both equations and I get $$\begin{align} mx&=\sqrt{1-(x-2)^2}\\ m&=\frac{\sqrt{1-(x-2)^2}}{x} \end{align}$$

$m$ can be put in the $y=mx$ equation, which equals to:
$$\begin{align} y&=\left(\frac{\sqrt{1-(x-2)^2}}{x}\right)x\\ &=\sqrt{1-(x-2)^2} \end{align}$$

But that equation has $\infty$ intersections, and I want only the equation who has $1$ interception.

What is the good way to know this? And how can it be calculated?

Answer

As you have in your post, we have $y = mx$ as the straight line. For this line to touch the semi-circle, we need that $y = mx$ and $y = \sqrt{1 - (x-2)^2}$ must have only one solution. This means that the equation $$mx = \sqrt{1 - (x-2)^2}$$ must have only one solution.

Hence, we need to find $m$ such that $m^2x^2 = 1 - (x-2)^2$ has only one solution.

$(m^2 + 1)x^2 -4x +4 = 1 \implies (m^2+1) x^2 - 4x + 3 = 0$.

Any quadratic equation always has two solution. The two solutions collapse to a single solution when the discriminant of the quadratic equation is $0$. This is seen from the following reasoning.

For instance, if we have $ax^2 + bx+c = 0$, then we get that $$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$$
i.e. $\displaystyle x_1 = \frac{-b + \sqrt{b^2 -4ac}}{2a}$ and $\displaystyle x_2 = \frac{-b - \sqrt{b^2 -4ac}}{2a}$ are the two solutions. If the two solutions to collapse into a single solution i.e. if $x_1 = x_2$, we get that $$\frac{-b + \sqrt{b^2 -4ac}}{2a} = \frac{-b - \sqrt{b^2 -4ac}}{2a}$$ This gives us that $\displaystyle \sqrt{b^2 -4ac} = 0$. $D = b^2 - 4ac$ is called the discriminant of the quadratic.

The discriminant of the quadratic equation, $(m^2+1) x^2 - 4x + 3 = 0$ is $D = (-4)^2 - 4 \times 3 \times (m^2+1)$.

Setting the discriminant to zero gives us that $(-4)^2 - 4 \times 3 \times (m^2 + 1) =0$ which gives us $\displaystyle m^2 + 1 = \frac43 \implies m = \pm \frac1{\sqrt{3}}$.

Hence, the two lines from origin that touch the circle are $y = \pm \dfrac{x}{\sqrt{3}}$.

Since you have a semi-circle, the only line that touches the circle is $\displaystyle y = \frac{x}{\sqrt{3}}$. (Thanks to @Joe Johnson 126 for pointing this out).