How does family of curves formula exactly work?

Question

I am familiar with the following equations

$$S_1 + \lambda S_2=0$$ for two circles

$$L_1 + \lambda L_2=0$$ for two lines $$L_1 + \lambda S_1=0$$ for circle and line

But I never really understood how they worked. Now this is a sample questions

A circles touches the parabola $y^2=2x$ at P $(\frac 12, 1)$ and cuts parabola at vertex V. If centre of circle is a Q, find the radius of circle

The formula used here was

$$(x-\frac 12)^2 + (y-1)^2 +\lambda (2x-2y+1)=0$$

Now it’s easy to see that equation is basically hinting at a curve passing through $(1/2, 1)$ and tangent to $(2x-2y+1)$, but how exactly was the form mat determined? How can we tell if this will us gives a circle? Why was the distance formula used in the first part of the equation? Basically I want to know the process of writing such equations .

Answer 1

$C_1:\left(x-\frac 12\right)^2 + (y-1)^2 =0$

is the equation of the circle having centre in $P\left(\frac12,1\right)$ and radius $r=0$ and $C_2:2x-2y+1=0$ is the equation of the line tangent to the parabola $\mathcal{C}:y^2=2x$ at $P$

A linear combination of the two $C_1+\lambda C_2=0$ represents all circles tangent to the parabola $\mathcal{C}$ at $P$ (see image (2) below) $$\left(x-\frac 12\right)^2 + (y-1)^2+\lambda(2x-2y+1)=0\tag{1}$$ If the circle passes through the origin, the vertex of $\mathcal{C}$, then substitute $(0,0)$ in $(1)$ to get $\lambda=-5/4$.

The circle we are looking for has equation $$\left(x-\frac{1}{2}\right)^2+(y-1)^2-\frac{5}{4} (2 x-2 y+1)=0$$ simplify $$2 x^2+2 y^2-7 x+y=0$$ center is $\left(\frac74,-\frac14\right)$ and radius $r=\frac{5\sqrt 2}{4}$

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Answer 2

If $S_1(x,y)=0$ and $S_2(x,y)=0$ are the equations of two curves, the equation

$$S_1(x,y)+\lambda S_2(x,y)=0$$ describes a family of curves that contains $S_1$ (when $\lambda=0$) and $S_2$ (when $\lambda=\infty$).

The type of the intermediate curves depend on the formula of the two curves. In particular, if the extreme curves are conics, so are all curves in the pencil. We can also say that if $S_1,S_2$ have common points (intersections), then any curve of the family passes through these points. (See why ?)