Limiting Point

Limiting points of a system of co-axial circles are the centres of the point circles belonging to the family (circles whose radii are zero are called point circle)

1. Limiting points of the co-axial system

Let the circle is ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+\mathrm{c}=0$

Where $\mathrm{g}$ is a variable and $\mathrm{c}$ is constant

$\therefore$ centre $(-\mathrm{g},0)$ and radius $\sqrt{{\mathrm{g}}^2-\mathrm{c}}$ respectively

Let $\sqrt{{\mathrm{g}}^2-\mathrm{c}}=0$ radius

${\mathrm{g}}^2-\mathrm{c}=0$

${\mathrm{g}}^2=\mathrm{c}$

$\mathrm{g}=\pm \sqrt{\mathrm{c}}$

Thus we get the two limiting points of the given co-axial system as

The limiting points are real and distinct, real and coincident or imaginary according as $\mathrm{C}>,=,<0$.

2. System of co-axial circles whose limiting points are given

Let $(\alpha ,\beta )$ and $(\gamma ,\delta )$ be the two given limiting points

Then corresponding circles with zero radii are

$(x-\alpha {)}^2+(y-\beta {)}^2=0=x^2+y^2-2\alpha x-2\beta y+{\alpha }^2+{\beta }^2=0$

$(\mathrm{x}-\gamma {)}^2+(\mathrm{y}-\delta {)}^2=0={\mathrm{x}}^2+{\mathrm{y}}^2-2\gamma \mathrm{x}-2\delta \mathrm{y}+{\gamma }^2+{\delta }^2=0$

System of co-axial circle equation is

$x^2+y^2-2\alpha x-2\beta y+{\alpha }^2+{\beta }^2+\lambda (x^2+y^2-2\gamma x+2\delta y+{\gamma }^2+{\delta }^2)=0(\lambda \ne -1)$

centre of this circle is $(\frac{\alpha +\gamma \lambda }{1+\lambda },\frac{\beta +\delta \lambda }{1+\lambda })$

and radius $=\sqrt{{\left(\frac{\alpha +\gamma \lambda }{1+\lambda }\right)}^2+{\left(\frac{\beta +\delta \lambda }{1+\lambda }\right)}^2-\frac{({\alpha }^2+{\beta }^2+\lambda {\gamma }^2+\lambda {\delta }^2)}{1+\lambda }}=0$

After solving find $\lambda$ substitute in (1)

We get the limiting point of co-axial system.

Properties of Limiting points

1. The limiting point of a system of co-axial circles are conjugate points with respect to any member of the system.

Let the equation of any circle be

${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+\mathrm{c}=0$

limiting points of (1) are $(\sqrt{\mathrm{c}},0),(-\sqrt{\mathrm{c}},0)$

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The polar of the point $(\sqrt{\mathrm{c}},0)$ is

$\mathrm{x}\sqrt{\mathrm{c}}+\mathrm{g}(\mathrm{x}+\sqrt{\mathrm{c}})+\mathrm{c}=0$

$(\mathrm{x}+\sqrt{\mathrm{c}})(\mathrm{g}+\sqrt{\mathrm{c}})=0$

$\mathrm{x}+\sqrt{\mathrm{c}}=0\Rightarrow \mathrm{x}=-\sqrt{\mathrm{c}}$

$\therefore (-\sqrt{c},0)$ ) lies on this.

Similarly $(\sqrt{\mathrm{c}},0)$ lies on polar with respect to $(-\sqrt{\mathrm{c}},0)$

$\therefore$ These limiting points $(-\sqrt{\mathrm{c}},0)$ and $(\sqrt{\mathrm{c}},0)$ are conjugate points

2. Every circle through the limiting points of a co-axial system is orthogonal to all circles of the system

Let the equation of any circle ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+\mathrm{c}=0$ where $\mathrm{g}$ is a variable and $\mathrm{c}$ is a constant Limiting point of this circle are $(-\sqrt{\mathrm{c}},0)$ and $(\sqrt{\mathrm{c}},0)$

Now let, ${\mathrm{x}}^2+{\mathrm{y}}^2+2{\mathrm{g}}_1\mathrm{x}+2{\mathrm{f}}_1\mathrm{y}+{\mathrm{c}}_1=$ be any circle passing through the limiting points $(-\sqrt{\mathrm{c}},0)$ and $(\sqrt{\mathrm{c}},0)$.

$\therefore \mathrm{c}-2{\mathrm{g}}_1\sqrt{\mathrm{c}}+{\mathrm{c}}_1=0$ (3) and $\mathrm{c}+2{\mathrm{g}}_1\sqrt{\mathrm{c}}+{\mathrm{c}}_1=0$

Solving (3) and (4). We get ${\mathrm{g}}_1=0$ and ${\mathrm{c}}_1=-\mathrm{c}$

$\therefore$ The equation becomes

${\mathrm{x}}^2+{\mathrm{y}}^2+2{\mathrm{f}}_1\mathrm{y}-\mathrm{c}=0$

applying condition of orthogonality

$2{\mathrm{gg}}_1+2{\mathrm{ff}}_1=\mathrm{c}+{\mathrm{c}}_1$ $\mathrm{o}+\mathrm{o}=\mathrm{c}-\mathrm{c}$

Hence condition is satisfied for all values of $g_1$ and $f_1$

Examples

1. If the origin be one limiting point of a system of co-axial circles of which $x^2+y^2+3x+4y+25=0$ is a member, find the other limiting point.

2. Find the radical axis of co-axial system of circles whose limiting points are $(-1,2)$ and $(2,3)$.

3. Find the equation of the circle which passes through the origin and belongs to the co-axial of circles whose limiting points are $(1,2)$ and $(4,3)$