P and Q are 4 units apart and on the same side of variable straight line which touches a circle

Question

My approach: I took a variable line ax+by+c=0 and took random points like (2,0) and (6,0). The I applied the formula for perpendicular distance and put it in the condition that PM+3QN=4. but this way there are three variable and one equation by which I am unable to get the equation of the line and satisfy it with the tangency condition of the circle that perpendicular distance from centre to this line is equal to radius but this way more and more number of variables are introduced and lesser number of equations.

Answer 1

For an arbitrary line $L$ extend $MN$ until it meets $PQ$ at point $A$. Introduce angle $\alpha=\angle MAP$ and length $b=AQ$. The following is obvious:

$$PM=(4+b)\sin\alpha$$

$$QN=b\sin\alpha$$

We know that $PM+3QN=4$. It means that:

$$(4+b)\sin\alpha+3b\sin\alpha=4$$

$$\sin\alpha=\frac1{b+1}\tag{1}$$

In other words, parameters $b$ and $\alpha$ are not independent. You can pick either $b$ or $\alpha$ freely and the other one can be calculated from (1).

Let us now calculate distance $d$ for an arbitrary point $R\in{PQ}$ such that $QR=x$:

$$d=(b+x)\sin\alpha=\frac{b+x}{b+1}\tag{2}$$

So for an arbitrary point $R$ and arbitrary line $L$, the distance from $R$ to $L$ is a function of $x$ (which represents position of point $R$ along the line $PQ$) and $b$ (which can be picked freely).

But there is one special point on segment PQ. If you put $x=1$ into (2) you get $d=1$ for all possible values of $b$. In other words, the point $R$ such that $PR=3,QR=1$ has the same distance $(d=1)$ from an arbitrary line $L$. So if you draw a circle around point $R$ with radius 1 it will be tangent to the line $L$.

It means that (A) and (B) are correct.

Answer 2

If you call $\theta$ the angle $\angle MPQ$, then you have $$\overline{PM} = 4-3\cdot\overline{QN} = 4\cos\theta-\overline{QN},$$ which gives you \begin{equation}\overline{QN} = 1-\cos\theta\tag{1}\label{uno}\end{equation} and consequently \begin{equation}\overline{PM} = 1+3\cos\theta.\tag{2}\label{due}\end{equation} Then it is useful to put $P$ in the origin of the axes.

Note that $$\overline{PH} = 3\cos\theta$$ and, thus, from \eqref{due}, $$\overline{HM} = 1.$$ Then if you draw the line $HK$ perpendicular to $PM$ you get $$\overline{KN} = 1,$$ and $$\overline{KQ} = \cos\theta.$$ Again, as in \eqref{uno}, $$\overline{QN} = 1-\cos\theta.$$ As a consequence, the distance of the line $MN$ from $G(3,0)$ will be constant, and the line will be tangent to the circle centered in $G$ with radius $1$.