Find the locus of the center of the circle for the following data

Let ABCD be a square of side length $$2$$ units. $$C_2$$ is the circle through vertices $$A,B,C,D$$ and $$C_1$$ is the circle touching all the sides of the square ABCD. $$L$$ is a line through $$A$$

A circle touches the line $$L$$ and the circle $$C_1$$ externally such that both the circles are on the same side of the line, then the locus of centre of the circle is

My attempt is as follows:-

Let the center of circle which touches the line $$L$$ and circle $$C_1$$ externally be $$C$$

$$CC_1=r+r_1\tag{1}$$

Here $$r$$ is the radius of circle with center $$C$$ and and $$r_1$$ is the radius of fixed circle $$C_1$$ with radius $$r_1$$

As $$L$$ is tangent to the circle $$CL=r\tag{2}$$

Dividing $$1$$ and $$2$$

$$\dfrac{CC_1}{CL}=1+\dfrac{r_1}{r}$$

Actual answer is parabola but in that case $$\dfrac{CC_1}{CL}=1$$ as eccentricity of parabola is $$1$$, but here $$1+\dfrac{r_1}{r}\ne1$$

Is my assumption of considering $$C_1$$ and $$L$$ as fixed circle and fixed line wrong? Please help me in this.

• yeah we can leave that, it has no importance. Feb 8 '20 at 9:48
• FYI - $C_1$ is the "inscribed" circle, and $C_2$ is the "circumscribed" circle. Feb 8 '20 at 15:24
• Your mistake is in assuming that the center of $C_1$ is the focus and $L$ is the directrix. Since the ratio of the distances is not constant, but varies with $r$, they cannot be the focus and directrix of a conic section. (That doesn't say the curve isn't a conic section - just that these two do not fit the roles of focus and directrix.) Feb 8 '20 at 15:39

Consider a line $$L'$$ parallel to $$L$$ and at a distance $$r_1$$ from $$L$$, such that $$CL'=r+r_1$$.